LIGHT 

A  TEXTBOOK  FOR  STUDENTS 

WHO  HAVE   HAD  ONE 

YEAR  OF  PHYSICS 


By 
H.  M.  REESE 

Professor  of  Physics  in  the 
University  of  Missouri 


(Columbia.  ffltBsmtri 

MISSOURI  BOOK  COMPANY 

1921 


COPYRIGHT  1921 
BY  THE  MISSOURI  BOOK  COMPANY. 


PREFACE. 

The  writing  of  this  book  was  undertaken  because  no  ex- 
isting text  on  the  subject  quite  filled  the  needs  for  my  own 
classes.  The  first  draft  was  mimeographed,  and  has  been  used 
in  that  form  with  some  success  for  several  years. 

It  is  planned  for  students  who  have  had  no  training  in 
the  calculus,  because  many  of  those  who  take  second-year 
physics  at  the  University  of  Missouri  suffer  from  that  handicap. 
The  first  part  has  purposely  been  made  rather  easy,  the  inten- 
tion being  to  lead  gradually  from  less  to  more  difficult  matter. 
A  persistent  attempt  is  made  to  lay  stress  upon  the  experimental 
basis  for  our  theories,  and  to  point  out  such  reasons  as  exist 
for  and  against  them;  because  my  own  experience  has  been 
that  many  students,  though  they  may  learn  the  facts  of  a 
science  conscientiously  and  in  a  sense  thoroughly,  fail  com- 
pletely to  realize  the  inductive  processes  on  which  the  theoreti- 
cal structure  is  founded,  thus  missing  one  of  the  chief  educa- 
tional values  to  be  derived  from  the  study  of  science.  It  is  in 
line  with  the  same  idea  that  certain  matters  have  been  intro- 
duced, particularly  in  the  last  two  chapters,  whose  purpose 
is  to  give  the  reader  an  idea,  incomplete;  though  it  may  be, 
of  the  present  state  of  optical  theory  and  allied  branches  of 
physical  science. 

Thanks  are  due  to  my  colleague,  Professor  0.  M.  Stewart, 
for  a  number  of  valuable  suggestions,  and  also  to  Professor 
Henry  Gale,  of  the  University  of  Chicago,  who  read  the 
manuscript  and  suggested  changes  and  additions  which  I  have 
been  glad  to  make. 

H.  M.  R. 

Columbia,  Missouri, 

October,  1920.  465254 


CONTENTS 

CHAPTER  I. 

Article  Page 

1.  Introduction 1 

2.  Velocity     3 

3.  Roemer's  method    4 

4.  Bradley's  method 5 

5.  Fizeau's  method  with  the  toothed  wheel 7 

6.  The  rotating  mirror  method 9 

CHAPTER  II. 

7.  Refraction  through  a  prism   15 

8.  Newton's  conception  of  color 18 

9.  Impure  colors   19 

10.  Color  due  to  absorption 20 

11.  Color  due  to  other  causes 22 

12.  Black  and  white 23 

13.  Complementary  colors  and  color  mixture 24 

14.  The  eye     26 

15.  Color  vision  theories    28 

CHAPTER  III. 

16.  The  corpuscular  theory  of  light 32 

17.  The  wave  theory   33 

18.  Bending  of  light  into  a  shadow 34 

19.  Nature  of  the  ether 36 

20.  Waves  in  general.     Plane  waves .  37 

21.  Mathematical  formula  for  a  wave 39 

22.  Interference.     Fresnel's  mirrors    41 

23.  Interference  in  white  light 48 

CHAPTER  IV. 

24.  Reflection  and  refraction.     Huyghens'  principle.     Index  of 
refraction    50 

25.  Total  reflection.     Critical  angle .  54 

26.  Deviation  through  a  prism   59 

CHAPTER  V. 

27.  Reflection   and   refraction   of   spherical   waves   at   a   plane 
surface 63 

vii 


Vlll  LIGHT 

Article  0  Page 

28.  Judgment  of  the  distance  of  an  image 67 

29.  Image  of  an  extended  object 69 

30.  Reflection  and  refraction  at  spherical  surfaces 69 

31.  Lenses 77 

32.  Two  lenses  in  contact 82 

33.  Chromatic    aberration 83 

34.  Achromatic   lenses .  84 

35.  Image  of  extended  object     Undeviated  ray 87 

36.  Magnification 88 

37.  Micrometer    90 

38.  Imperfections    of   mirrors    and    lenses    91 

39.  Spherical  aberration    91 

40.  Curvature  of  field 92 

41.  Astigmatism     93 

42.  Lenses  for  special  purposes 94 

CHAPTER  VI. 

43.  The    telescope 97 

44.  Magnifying  power 98 

45.  Ramsden   eyepiece    100 

46.  Opera  glass 101 

47.  Prism   binocular    •  .  .  .  .  103 

48.  Reflecting  telescopes 103 

49.  Simple    microscope     104 

50.  Compound  microscope    105 

51.  Projection  lanterns 107 

CHAPTER  VII. 

52.  Prism    spectroscope    110 

53.  Brightline   spectra Ill 

54.  Spectral   series    114 

55.  Continuous   spectra    116 

56-  Dark-line  spectra    117 

57.  Absorption  by  solids  and  liquids 118 

58.  Continuous  spectrum  of  an  absolutely  black  body 119 

59.  Planck's  theory  of  "quanta"    120 

60.  The  plane  grating 121 

61.  Why  the  lines  are  sharp 126 

62.  Reflection    gratings 129 

63.  The  concave  grating   129 

64.  The    ultraviolet    region.     Fluorescence.     Phosphorescence. 
Photography 130 

65.  The  infrared  region   131 

66.  The   bolometer    132 

67'.  The    thermopile     132 

68.  The  Doppler  principle.     Motion  of  the  stars 133 


CONTENTS  I 

CHAPTER  VIII. 

Article  Page 

69.  The  approximately  rectilinear  propagation  of  light 138 

70.  Shadow  of  an  edge    142 

71.  Shadow  of  a  wire 145 

72.  Diffraction  through  a  rectangular  opening    145 

73.  Resolving-power    148 

CHAPTER  IX. 

74.  Young's  interference   experiment    152 

75.  The   biprism    152 

76.  Interference  in  thin  uniform  films 153 

77.  Change  of  phase  on  reflection 155 

78.  Non-uniform  films    159 

79.  The  Michelson  interferometer 160 

80.  Newton's  rings    162 

81.  Fabry  and  Perot  interferometer 163 

82.  Interference  in  white  light  .  .  . 163 

83.  Rainbows 167 

84.  Motion  relative  to  the  ether    171 

85.  The   relativity  theory    ' 173 

CHAPTER  X. 

86.  Simple  harmonic  motion 175 

87.  Velocity  in   S.   H.   M 178 

88.  Acceleration  in  S.  H.  M 180 

89.  Energy  in  S.  H.  M 182 

90.  Two  parallel  S.  H.  M.'s    184 

91.  Application  to  cases  of  interference 187 

92.  Two  S.  H.  M.'s  at  right-angles   190 

93.  Lissajous   figures    193 

CHAPTER  XI. 

94.  Inverse  square  law    195 

95.  Photometry    196 

96.  Rumford  photometer   196 

97.  Bunsen   photometer    197 

98.  Lummer-Brodhun  photometer 198 

99.  Light-standards 199 

100.  Solid  angle    199 

101.  Intrinsic  luminosity    200 

102.  Spectrophotometer    204 


X  LIGHT 

CHAPTER  XII. 

Article  Page 

103.  Transverse   and   longitudinal   waves    205 

104.  Double  refraction 206 

105.  Polarization  of  the  O  and  E  light 209 

106.  Wave-surface  in  doubly-refracting  crystals    213 

107.  The  lateral  displacement  of  the  E-ray    213 

108.  Special  cases  of  double  refraction 215 

109.  Tourmaline 216 

110.  Biaxial    crystals    217 

111.  Polarization  by  reflection    •. 218 

CHAPTER  XIII. 

112.  Methods    of    polarizing   light    223 

113.  The   Nicol   prism    224 

114.  Double-image    prisms     225 

115.  Crossed  Nicols  and   crystal   plate    226 

116.  Elliptic  polarization    226 

117.  Circular  polarization ". 228 

118.  Rotation  of  the  plane  of  polarization 230 

119.  Magnetic  rotation 233 

120.  The  rings-and-brushes  phenomenon    234 

121.  The  nature  of  elliptic  and  circular  polarization    234 

CHAPTER  XIV. 

122.  Plane  of  polarization  and  plane  of  vibration 240 

123.  Elastic-solid  theories    240 

124.  Electromagnetic  theory   241 

125.  Direction  of  the  vibrations 243 

126.  Fundamental   electromagnetic  laws    245 

127.  Faraday's  displacement-currents 245 

128.  Maxwell's    assumption    -  247 

129-  Hertz's  experiments   .  ., 247 

130.  Propagation  of  electromagnetic  waves    248 

131.  Velocity  of  the  waves   250 

132.  Refractive  index  and  dielectric  constant    .                             .  252 


CHAPTER  XV. 

133.  Dispersion    254 

134.  Electron  theory  of  matter 255 

135.  Electromagnetic   dispersion   formula    255 

136.  Anomalous   dispersion    257 

137.  Reststrahlen  .  259 


CONTEXTS  XI 

CHAPTER  XVI. 

Article  Page 

138.  Production   of  X-rays 262 

139.  Their  properties   263 

140.  Are  X-rays  ether  waves? 264 

141.  Crystal  reflection  of  X-rays 266 

142.  Measurement  of  wavelengths.     Crystal  structure 268 

143.  X-ray  spectra    270 

144.  The  K  and  L  series 270 

145.  Quantum  theory  applied  to  X-rays 271 

146.  Secondary  X-rays.     Absorption    271 

147.  Total  range  of  ether  waves 272 

CHAPTER  XVII. 

148.  Review  of  the  development  of  light-theory   273 

149.  Modern  lines  of  investigation    274 

150.  The  Zeeman  effect 275 

151.  Lorentz's   theory    276 

152.  The   Stark   effect    278 

153.  The  photo-electric  effect   278 

154.  Atom-models    279 

155.  Bohr's  theory  of  the  hydrogen  atom 282 

Appendix  I.  Wavelengths  for  laboratory  use 286 

Appendix  II.    The  velocity  of  electromagnetic  waves   287 

Index                                                                                                           .  291 


LIGHT. 

CHAPTER  I. 

1.  Introduction. — 2.  Velocity.— 3.  Roemer's  method. — I.  Bradley's 
method. — 5.  Fizeau's  method  with  the  toothed  wheel. — 6.  The  rotating 
mirror  method  of  Foucault  (and  Fizeau). 

1.  Introduction. — Simple  and  familiar  observations  teach 
us  that  the  sensation  of  vision  is  caused  by  some  agency  that 
emanates  from  bodies  external  to  us,  and  enters  our  eyes.  For 
instance,  we  cannot  see  anything  in  a  room  with  tightly  closed 
blinds  where  there  is  no  source  of  artificial  illumination,  such 
as  a  candle,  fire,  or  electric  lamp.  Incidently,  this  experience 
also  teaches  that  we  may  classify  the  objects  we  see  into  two 
groups:  first,  luminous  objects,  such  as  a  candle,  a  fire,  the 
sun,  or  the  stars;  second,  objects  such  as  a  book,  a  tree,  the 
walls  of  the  room,  etc.,  which  can  be  seen  only  through  the 
agency  of  a  luminous  body. 

From  the  physicist's  point  of  view,  the  study  of  light  is 
the  study  of  this  activity,  whatever  its  nature  may  be,  which 
originates  in  luminous  bodies  and  causes  the  sensation  of 
vision  when  it  enters  the  eye.  His  interest  lies  primarily  in 
the  way  this  agency  starts  into  action  in  a  luminous  object, 
how  it  propels  itself  through  space,  how  it  behaves  on  striking 
objects  of  different  kinds,  such  as  glass,  crystals,  silver,  water, 
etc.,  and  its  relations  to  all  other  physical  phenomena,  such  as 
heat,  electricity,  and  magnetism. 

On  the  other  hand,  the  physicist  proper  does  not  concern 
himself  much  with  the  parts  that  the  eye  and  the  nervous 
system  play,  in  registering  in  our  consciousness  the  sensation 
(vision),  whose  primary  cause  is  the  physical  agency  that  we 
call  light.  This  question  is  important  and  interesting  enough, 
but  it  belongs  primarily  to  the  domains  of  the  physiologist 
and  the  psychologist. 

Let  us  begin  our  study  by  making  a  summary  of  such 
facts  as  common  knowledge  gives  us  about  light. 

In  the  first  place,  besides  differences  in  brightness,  which 
may  be  called  a  matter  of  quantity,  there  are  also  differences 
of  quality  to  be  considered,  as  shown  in  the  phenomena  of 
color. 

1 


2  LIGHT 

Second,  light  travels  approximately  in  straight  lines,  as 
is  shown  in  the  formation  of  shadows.  Nevertheless,  we  shall 
see  later  that  light  does  bend  around  corners  to  some  slight 
extent,  though  not  nearly  so  much  as  sound  does. 

Third,  it  differs  from  sound  also  in  that  it  travels  with- 
out hindrance  through  a  vacuum.  In  coming  to  us  from  the 
stars,  it  travels  through  millions  of  miles  of  the  most  perfectly 
empty  space  obtainable. 

Fourth,  it  is  either  itself  a  manifestation  of  energy,  or 
else  it  carries  energy  with  it,  since  any  object  which  receives 
and  absorbs  it  becomes  heated. 

Fifth,  when  it  strikes  a  surface,  more  or  less  of  it  is  gen- 
erally reflected.  (Those  exceptional  surfaces  which  reflect  no 
light  are  said  to  be  black).  If  the  surface  is  highly  polished, 
the  light  is  reflected  at  a  definite  angle,  in  which  case  we  say 
the  reflection  is  regular.  If  the  surface  is  rough,  like  that  of 
a  sheet  of  paper,  the  light  is  scattered  in  all  conceivable  direc- 
tions, and  the  reflection  is  said  to  be  diffuse. 

Sixth,  there  are  many  substances,  such  that  when  light 
strikes  their  surfaces,  although  part  is  reflected,  part  enters 
the  material  and  passes  through  it  rather  freely.  Such  ma- 
terials are  said  to  be  transparent.  Light  traverses  transparent 
materials  approximately  in  straight  lines,  as  it  does  the  air 
or  free  space,  but  there  is  an  abrupt  bending  of  the  rays  at 
the  place  where  they  pass  through  the  surface.  This  bending 
is  called  refraction. 

Seventh,  light  travels  either  instantaneously,  or  else  with 
enormous  velocity.  Here  again  the  comparison  with  sound  is 
very  striking.  The  phenomenon  of  echoes  shows  that  sound 
travels  with  a  speed  which,  though  great,  cannot  be  called 
enormous,  and  indeed  a  fairly  accurate  measurement  of  this 
speed  could  be  made  by  noting  with  a  stop-watch  the  time 
required  for  an  echo  to  be  heard  from  a  cliff  or  large  building, 
whose  distance  from  the  observer  is  known.  An  exactly 
analogous  experiment  with  light  would  be  to  note  with  a  stop- 
watch the  time  that  elapses  between  the  flashing  of  a  light  and 
the  perception  of  its  reflection  in  a  mirror,  whose  distance  from 
the  observer  is  known.  Such  an  experiment  would  fail  com- 
pletely, because  no  stop-watch  could  record  a  short  enough 


VELOCITY  3 

time-interval;  and  even  without  that  objection,  no  human 
being  has  a  "reaction-time"  constant  enough  to  manipulate 
a  stop-watch  with  anything  like  the  necessary  precision. 

2.  Velocity. — Of  the  above  mentioned  seven  points  of 
common  knowledge  about  light,  the  last  (in  regard  to  velocity) 
is  of  so  much  interest,  and  can  be  so  easily  discussed  without 
a  thorough  knowledge  of  other  optical  phenomena,  that  we 
shall  consider  it  here  at  some  length. 

It  is  interesting  to  note  that  Galileo  actually  tried  to 
measure  the  velocity  of  light  by  the  method  outlined  above, 
except  that  instead  of  using  a  mirror  to  send  back  the  light 
(probably  none  then  available  were  good  enough  to  use  over 
great  distances)  he  stationed  two  observers  with  lanterns  a 
great  distance  apart.  Observer  number  one  flashed  his  lantern, 
and  number  two  answered  by  a  flash  of  his  own  as  quickly  as 
possible.  Number  one  then  tried  to  measure  the  interval  of 
time  between  his  own  signal  and  his  perception  of  the  answer- 
ing signal.  Of  course  no  perceptible  time-interval  was  found, 
and  Galileo  concluded  from  this  that  the  velocity  of  light  was 
too  great  to  measure. 

Since  a  velocity  is  always  a  distance  divided  by  a  time, 
Galileo's  failure  shows  that  in  order  to  measure  so  great  a 
velocity  we  may  proceed  in  one  of  three  possible  ways.  First, 
we  may  choose  a  distance  so  great  that,  in  spite  of  the  great 
velocity  to  be  measured,  the  interval  of  time  will  be  large 
enough  to  measure  conveniently  by  ordinary  methods.  Second, 
it  might  be  possible  to  get  a  direct  comparison  between  the 
velocity  of  light  and  some  known  velocity  (such  as  that  of 
the  earth  in  its,  orbit)  which,  although  much  smaller,  is  yet 
far  greater  than  that  of  anything  we  can  handle  in  the  labora- 
tory. Third,  we  may  return  to  the  principle  of  Galileo's 
method,  with  a  relatively  short  distance  (say  a  few  miles)  and 
correspondingly  small  time-interval,  if  we  use  a  mirror  to  re- 
turn the  light  and  find  some  very  refined  method  far  measur- 
ing an  exceedingly  short  time.  The  last  method  would  have 
this  advantage  over  the  other  two,  that  since  the  distance 
concerned  is  not  excessive,  it  might  be  possible  to  measure  the 
velocity,  not  only  in  air  or  in  free  space,  but  also  in  water  and 
other  transparent  materials. 


4  LIGHT 

It  is  a  matter  of  historical  fact  that  each  of  the  three 
possibilities  suggested  above  has  been  successfully  carried  out, 
— the  first  !by  the  Danish  astronomer  Roemer,  in  1676,  the 
second  by  the  English  astronomer  Bradley,  in  1728,  and  the 
third  by  two  French  physicists,  Fizeau  and  Foucault,  in  1849 
and  1850  respectively. 

3.  Roemer 's  Method. — The  planet  Jupiter,  like  the  earth, 
revolves  about  the  sun  in  a  nearly  circular  orbit,  but  its  orbital 
radius  is  so  much  larger  that  it  takes  nearly  twelve  of  our 
years  to  complete  the  circuit.  It  has  several  satellites,  similar 
to  our  moon,  one  of  which  circles  the  planet  in  about  11 
hours.  Once  in  every  revolution,  it  enters  the  shadow  of 
Jupiter  and,  since  it  is  not  a  luminous  body,  but  can  be  seen 
only  by  virtue  of  the  sun's  light,  it  then  disappears  for  a  short 
time.  The  interval  of  time  between  two  successive  eclipses  is 
called  the  period.  We  would  naturally  expect  the  period  to  be 
constant,  but  it  was  long  known  that  it  seems  to  vary,  accord- 
ing to  the  relative  positions  of  Jupiter  and  the  earth.  In 
figure  1,  the  larger  circle  represents  the  orbit  of  Jupiter,  the 

^+.~ -^  smaller  that  of  the  earth,  with  the 

/'  sun  at  the  center  of  both.    (Actu- 

/  \     ally  each  orbit  is  an  ellipse,  with 

"V  B^  \   the  sun  at  one   focus.)      Suppose 

I  J&  (  V  c?   *E,  1  that  at  a  given  time  the  earth  is 

I    \  &^..''  i  at  Ej  and  Jupiter  at  J1?  the  two 

**N  '    being    in    line    with    the    sun.     A 

»  / 

\  /      little  more  than  six  months  later, 

NVV^  ^/  they  will   again   be   in  line,   with 

the   sun   between  them,   the   earth 

at  E2,  Jupiter  at  J2,  for  Jupiter 

moves  more  slowly  in  its  orbit  than  does  the  earth.  Again,  at  a 
still  later  time,  the  earth  will  be  at  E3  and  Jupiter  at  J3,  the 
former  having  made  something  more  than  a  complete  circuit, 
while  the  latter  has  travelled  only  through  the  arc  J^Js- 
Evidently  there  are  times,  as  at  A,  when  we  are  receding 
from  Jupiter,  and  other  times,  as  at  B,  when  we  are  approach- 
ing him.  It  was  noticed  that  when  the  earth  is  receding  from 
Jupiter  the  period  of  the  satellite  seems1  to  be  longer,  when 
it  is  approaching  him  shorter,  than  the  average.  Thus  if, 


BRADLEY 'S  METHOD  5 

when  the  earth  is  in  such  a  position  as  Ex,  with  Jupiter  at  J\, 
a  complete  schedule  of  satellite  eclipses  be  made  out  in  ad- 
vance, on  the  supposition  that  they  occur  with  a  regular  period, 
it  will  be  found  that  they  appear  more  and  more  behind 
schedule  time,  till  the  earth  and  Jupiter  are  in  the  positions 
E2  J2,  and  then  begin  to  pick  up  till  they  are  again  actually 
on  schedule  time,  when  the  earth  is  at  E3  and  Jupiter  at  J3. 

Roemer  saw  that  this  phenomenon  could  be  explained 
perfectly  by  supposing  that  the  eclipses  occur  at  perfectly 
regular  intervals,  provided  that  a  finite  time  is  required  for 
the  light  which  brings  us  the  news  of  an  eclipse  to  travel  the 
very  great  distances  involved.  For  when  we  are  moving  away 
from  Jupiter,  as  at  the  position  A,  each  succeeding  eclipse  is 
announced  to  us  by  light  that  must  travel  a  somewhat  greater 
distance,  and  therefore  the  apparent  period  would  be  increased 
by  the  time  required  for  the  light  to  travel  the  additional 
distance.  On  the  basis  of  a  schedule  of  eclipses,  such  as  was 
described  in  the  previous  paragraph,  it  is  found  that  the 
eclipses  observed  when  the  earth  and  Jupiter  are  on  opposite 
sides  of  the  sun  seem  to  be  about  16.6  minutes  behind  the 
schedule.  According  to  Roemer 's  views,  this  would  indicate 
that  it  takes  that  much  time  for  light  to  cross  the  earth's 
orbit.  Since  the  mean  radius  of  the  orbit  is  about  92.8  X  10<J 
miles,  this  gives  for  the  velocity  of  light  18.6  X  104  miles/sec., 
which  is  the  same  as  2.99  X  1010  cm./sec. 

It  is  worth  noticing  that  the  great  distance  of  Jupiter 
from  the  earth  does  not  enter  into  the  problem,  only  the 
changes  in  that  distance,  and  it  would  not  be  possible  to  deter- 
mine the  velocity  of  light  by  observations  of  the  satellite  if  the 
earth  were  stationary  with  respect  to  the  planet.  It  is  true 
that  each  eclipse  would  be  observed  some  time  later  than  its 
actual  occurrence,  but  we  could  not  know  how  much  later, 
unless  we  already  knew  the  velocity  as  well  as  the  distance 
from  Jupiter,  and  calculated  back  from  the  time  of  the  appear- 
ance to  the  time  of  actual  occurrence.  That  is,  one  would  have 
to  know  the  very  thing  which  it  is  his  object  to  find. 

4.  Bradley 's  Method. — The  so-called  fixed  stars  are  so  far 
from  us  that  the  relatively  small  range  of  motion  of  the  earth 
in  its  orbit  hardly  changes  their  apparent  positions,  that  is. 
their  directions  from  us.  Still,  as  astronomical  methods  became 


LIGHT 


more  refined,  it  was  observed  that  when  the  earth  is  on  one 
side  of  its  orbit  the  position  of  a  star  seems  shifted  slightly 
to  the  other  side,  as  we  should  expect  from  ordinary  geometry. 
Of  course  this  effect,  which  we  call  "parallax,"  is  most  pro- 
nounced on  the  nearest  stars,  and  its  measurement  enables  us 
to  estimate  the  distance  of  such  stars. 

The  phenomenon  discovered  by  Bradley,  known  as  "aber- 
ration," is  an  entirely  different  matter,  though  it  too  consists 
of  an  apparent  change  in  the  position  of  a 
star.  It  is  a  shift,  not  in  a  direction  opposite 
to  that  in  which  the  earth  stands  from  the 
sun,  but  towards  the  direction  in  which  the 
Dearth  is  moving  at  the  time,  that  is,  it  de- 
pends not  on  the  position  of  the  earth,  but  on 
its  velocity.  Bradley  found  an  explanation 
for  this  phenomenon,  suggested  by  the  way 
a  flag  acts  when  it  is  affected  both  by  the 
wind  and  by  the  motion  of  the  ship  on  which 
it  is  carried.  For  instance,  in  figure  2,  let  the 
vector  v  represent  the  velocity  of  the  ship, 
V  that  of  the  wind,  which  is  here  supposed 
to  blow  directly  across  the  ship.  The  flag  will  not  stand  out 
in  the  true  direction  of  the  wind,  but  in  a  direction  such  as 
AB.  That  is,  the  flag  is  affected  not  only  by  the  true? 
wind,  but  also  by  an  apparent  wind  equal  and  opposite  to  the 
velocity  of  the  ship.  Evidently  it  will  form  an  angle  with 
the  true  direction  of  the  wind  whose  tangent  is  v/V. 
This  comes  to  the  same  thing  as  saying  that  to  a 
person  travelling  with  the  ship  the  wind  appears  to 
come,  not  from  the  true  direction  M,  but  from  the 
direction  N,  where  tan.  a  —  v/V. 

Now  let  us  take  another  figure  (3)  in  which 
we  replace  the  ship  by  the  earth  moving  through 
space,  and  the  wind  by  light  coming  from  a  star 
whose  position  is  broadside  on  to  the  earth's  mo- 
tion. Instead  of  the  velocity  V  of  the  wind,  we 
have  the  velocity  c,  of  light.  Then  the  star,  whose 
real  position  is  in  the  direction  M,  will  appear  in 
the  direction  N,  making  an  angle  a  with  the  true  direction, 
such  that  tan.  a  —  v/c.  The  angle  a  cannot  be  measured 


Figure    2 


Figure    3 


THE  TOOTHED  WHEEL  METHOD 


directly,  since  we  see  only  the  apparent  direction  of  the 
star,  not  its  true  position;  and  we  could  never  hope  to  find 
the  velocity  by  means  of  the  aberration  if  the  earth  continued 
alwaj^s  moving  in  the  same  direction.  But  six  months  later  the 
earth  would  be  on  the  opposite  side  of  its  orbit,  and  moving  on 
the  opposite  direction.  Therefore,  if  we  take  the  angle  between 
the  two  apparent  positions  of  a  star  at  times  six  months  apart, 
this  will  be  twice  the  angle  a.  Then  knowing  the  velocity  of 
the  earth  in  its  orbit,  we  can  at  once  calculate  the  velocity 
of  light.  This  was  done  by  Bradley,  who  thus  got  a  value 
for  c  which  was  quite  close  to  that  obtained  by  Roemer's 
method. 

5.  Fizeau's  Method  with  the  Toothed  Wheel. — The  main 
idea  in  this  method  is  to  send  a  beam  of  light  through  a  small 
hole  toward  a  mirror  from  which  it  is  reflected  back  toward 
the  hole  The  hole  is 
opened  and  closed 
very  rapidly,  the  ra- 
pidity of  this  action 
being  gradually  in-  gl 
creased  till  the  light  ly 
that  passed  through 
the  opening  going  out 
finds  it  closed  when  it  returns.  This  rapid  opening  and  closing 
of  an  aperture  was  accomplished  by  using  a  toothed  wheel,  as 
shown  in  profile  in  figure  4.  The  wheel  was  rotating  rapidly  at 

a  controlled  speed, 
and  the  light  passed 
out  through  the 
gaps  between  the 
teeth.  The  actual 
arrangement,  of  the 
optical  parts  of  the 
apparatus  was  much 
more  c  o  mplicated 
than  the  simple  dia- 
gram of  fi  g  u  r  e  4, 
and  is  better  shown 

in  figure  5,  which  we  shall  consider  in  detail  because  it  involves 
so  many  of  the  principles  common  to  most  optical  experiments. 


SOURCE 


Figure    4 


K 


8  LIGHT 

In  the  first  place,  since  an  observer  must  watch  to  see 
when  the  proper  speed  is  reached  to  prevent  the  light  from 
returning  through  the  hole  through  which  it  passed  out,  some- 
place must  be  left  for  the  eye,  which  obviously  cannot  be 
placed  either  in  front  of  the  source  of  light  or  behind  it.  The 
source,  for  instance  an  arc-light,  is  therefore  placed  to  one 
side,  as  at  S,  and  reflected  toward  the  wheel  by  a  small  mirror 
MM,  placed  at  45°.  This  is  not  a  heavily  silvered  affair,  like 
a  household  mirror,  but  either  a  simple  unsilvered  sheet  of 
plane  glass,  or,  better  yet,  .one  which  has  a  coating  of  silver 
so  thin  that  about  half  the  light  is  reflected,  half  transmitted. 
It  is  obvious  that  by  such  a  device  half  the  available  light  is 
lost,  by  passing  through  in  the  direction  of  K,  and  half  of 
what  is  left  is  again  lost  in  the  returning  beam,  for  it  also 
is  half  reflected,  so  that  only  %,  or  even  less,  of  the  original 
beam  can  reach  the  eye  placed  at  E.  Nevertheless,  enough  is 
left  for  the  purpose,  and  it  is  possible  with  this  arrangement 
to  see  the  returning  light  without  interfering  with  its  passage 
outward. 

Besides  the  mirror,  a  system  of  lenses  is  introduced,  whose 
purpose  is  two-fold:  first  to  prevent  the  light  from  spreading 
out  indefinitely,  and  so  becoming  weakened,  second  to  concen- 
trate it  Avhere  it  meets  the  rim  of  the  wheel,  so  that  it  can  pass 
through  only  one  gap  at  a  time.  The  lens  L1?  between  the 
source  and  the  inclined  mirror,  forms  an  image  of  the  source 
just  at  the  point  where  it  passes  through  the  wheel.  If  one  of 
the  openings  happens  to  be  at  this  point  the  light  will,  pass 
through,  spreading  out  from  the  hole  just  as  if  the  latter  were 
itself  the  source.  A  second  lens  L2,  whose  principal  focus  is 
at  the  rim  of  the  wheel,  receives  the  rays  and  converts  them 
into  a  parallel  beam,  which  can  travel  to  any  required  distance 
without  suffering  any  further  weakening-  except  that  due  to  the 
small  and  unavoidable  absorption  by  the  air  through  which  it 
passes.  In  Fizeau's  experiment  it  was  carried  about  4  miles. 
At  the  end  of  this  distance,  it  could  be  made  to  fall  upon  a 
plane  mirror,  as  shown  in  the  diagram  of  figure  4,  and  so  re- 
flected back,  but  a  plane  mirror  perfect  enough  over  its  whole 
surface  for  this  purpose  would  be  hard  to  construct.  Conse- 
quently, a  third  lens,  L3,  is  inserted,  at  whose  principal  focus 


THE  ROTATING  MIRROR  METHOD  9 

is  a  concave  mirror  PP,  with  its  center  of  curvature  at  the 
lens.  By  this  means  only  a  small  portion  of  the  mirror  is 
used,  and  it  makes  little  difference  if  it  is  not  perfect  all  over. 

The  light  now  returns  through  L3  and  L2,  is  again  focussed 
on  the  same  part  of  the  wheel,  and  if  an  opening  is  there  it 
passes  through,  part  of  it  getting  through  the  mirror  MM  to 
the  eye. 

Now  consider  what  happens  as  the  wheel  is  started  in 
revolution,  slowly  at  first,  but  with  increasing  speed.  At  first 
a  flash  will  be  seen  whenever  a  gap  in  the  wheel  leaves  the 
path  clear.  But  the  eye  cannot  detect  flashes  coming  more 
frequently  than  about  20  per  second,  therefore  the  succession 
of  flashes  will  change  gradually  to  an  apparently  steady  light 
as  the  speed  of  the  wheel  increases,  for  much  more  than  20 
teeth  per  second  must  pass  across  the  field  of  vision  before  the 
speed  is  sufficient  for  a  tooth  to  completely  block  the  path  of 
the  returning  ray  which  went  out  through  the  adjacent  gap. 
But  if  the  speed  of  rotation  is  increased  still  more,  this  appar- 
ently steady  light  will  gradually  become  fainter,  as  each  tooth 
encroaches  more  and  more  on  the  returning  beam.  It  vanishes 
completely  when  the  speed  is  such  that  a  tooth  moves  into  the 
place  formerly  occupied  by  a  gap  while  the  light  is  passing 
from  the  wheel  to  the  concave  mirror  and  back  again.  It  is 
obvious  that  a  further  increase  of  speed  will  cause  a  reappear- 
ance of  the  light,  which  indeed  can  go  through  a  number  of 
maxima  of  intensity,  separated  by  total  darkness,  if  the  speed 
of  rotation  increases  indefinitely. 

In  this  experiment,  the  toothed  wheel,  whose  speed  of 
rotation  can  be  determined  by  suitable  mechanical  devices, 
serves  as  a  means  of  measuring  very  short  time-intervals,  thus 
enabling  us  to  measure  the  velocity  of  light  over  relatively 
short  distances. 

6.  The  Rotating  Mirror  Method  of  Foucault  (and  Fizeau). 
— This  method,  first  proposed  by  Arago,  another  Frenchman, 
was  worked  out  by  Foucault  and  Fizeau,  jointly  at  first,  but 
afterwards  independently.  Foucault  finished  the  task  first, 
Fizeau  having  been  delayed  by  an  accident. 

In  this  case,  as  in  that  of  the  toothed-wheel  method,  we 
shall  consider  first  a  diagram  showing  only  the  crude  principle 


10  LIGHT 

of  the  method,  figure  6.  Mt  is  a  flat  mirror,  which  can  be 
rotated  rapidly  about  an  axis  in  its  own  plane,  perpendicular 
to  the  plane  of  the  paper.  It  receives  a  beam  of  light  from  a 
source  S,  and  reflects  it  in  a  direction  depending  upon  the 
po'sition  of  the  mirror  at  the  instant.  Once  in  every  revolu- 
tion, and  only  during  a  very  small  part  of  the  revolution,  the 
reflected  light  falls  upon  a  second  mirror,  M2,  which  reflects 
it  back  to  M1?  and  Mt  in  turn  reflects  it  back  toward  the 

source.  If  the  mirror  were  sta- 
tionary, or  if  the  velocity  of 
light  were  infinite,  so  that  it 
travelled  from  Mx  to  M2  and 
back  before  Ma  had  turned  at 
all,  then  the  returning  beam 
would  come  exactly  back  to  the 
source  S.  .  But  since  it  requires 
a  finite  time  to  traverse  this  dis- 
tance, the  mirror  will  have 
turned  a  little,  and  the  beam 
Figure  6  will  not  return  exactly  to  the 

source,  but  to  a  new  point  S'. 

Here,  as  in  the  tooth-wheel  method,  some  other  details 
must  be  introduced  to  make  the  experiment  really  possible. 
The  most  important  thing  to  remember  in  all  optical  experi- 
ments is  that  a  beam  of  light  never  consists  of  a  single  ray, 
but  always  of  a  great  many,  generally  having  different 
directions.  For  this  reason  it  is  impossible  to  make  accurate 
measurements  of  position  unless,  by  the  use  of  lenses  or  other 
means,  the  rays  are  concentrated  to  a  definite  focus. 

Figure  7  is  a  complete  diagram  of  the  apparatus.  Light 
from  the  source  S  passes  through  a  narrow  slit,  introduced  to 
give  sharp  edges  to  the  illuminated  area,  and  falls,  after  re- 
flection from  the  haM-silvered  45-degree  mirror  mm,  upon  the 
lens  L,  which  would,  if  the  mirror  Mt  were  not  present,  bring 
it  to  a  focus  at  the  point  Ix.  The  mirror  M2  is  concave,  form- 
ing a  section  of  a  sphere  whose  center  is  at  the  axis  of  M^ 
and  which  passes  through  the  point  Ilm  Let  R  be  the  radius 
of  this  sphere,  that  is  the  distance  from  Mt  to  M2.  As  the 
mirror  Mt  rotates,  the  direction  of  the  rays  that  it  reflects 


THE  ROTATING  MIRROR  METHOD  11 

changes,  but  always  in  such  a  way  that  an  image  of  the 
illuminated  slit  is  formed  somewhere  on  the  circle  I^MoB. 
We  may  regard  this  image  as  sweeping  round  the  circle  with 
an  angular  velocity  twice  that  of  the  rotating  mirror.  During 
a  short  interval  of  time,  it  will  fall  upon  the  fixed  mirror  M2, 
which  reflects  the  light  directly  back  over  its  original  path  as 


-->         -_--—----      -- 


B 
Figure    7 

far  as  M,.  If  the  latter  had  been  at  rest,  the  light  would 
have  continued  to  retrace  its  path  as  far  as  the  half-silvered 
mirror  mm.  Part  of  it  would  then  penetrate  mm  and  come 
to  a  focus  at  fx,  forming  there  an  image  that  could  be  seen  by 
the  eye  at  E,  aided  perhaps  by  an  eye-lens  1.  It  will  be 
noticed  that  this  returning  light  would  act  exactly  as  if  it 
came  from  the  point  Ia  instead  of  from  M2.  In  fact,  It  is  the 
"image  by  reflection"  of  the  point  M2,  formed  by  the  mirror 
Mt.  According  to  the  laws  of  reflection  in  a  plane  mirror, 
which  will  be  taken  up  in  detail  later,  the  angles  A1M1M2  and 
AJM^I!  are  equal,  Al  being  the  point  where  the  plane  of  the 
mirror  Mt  cuts  the  circle.  Since  the  rays  striking  the  lens 
would  appear  to  come  from  I,,  the  image  fx  would  be  the 
;' conjugate  focus"  of  Ix.  According  to  the  law  of  lenses,  the 


12  LIGHT  . 

line  I1f1  passes  through  the  center  of  the  lens,  and  fx  is  at 
such  a  distance  that 

— +— =  ! 

Lfj,      LIt      F 

where  F  is  what  we  call  the  "focal  length"  of  the  lens  L. 
(section  31) 

In  point  of  fact,  however,  M0  instead  of  being  at  rest, 
is  rotating  in  the  direction  shown  by  the  arrow.  Consider  the 
light  that  starts  from  it  toward  M,  when  Mt  is  in  the  position 
indicated.  When  this  light  returns  to  Mt  the  latter  will  have 
turned  through  a  small  angle  a,  so  that  its  plane  will  now 
intersect  the  large  circle  in  the  point  A2.  Therefore  it  will 
reflect  the  light  returned  from  M2  as  if  it  came,  not  from  I±, 
but  from  a  new  point  I2,  such  that  A2M1I2  =  A2M1M2.  After 
reflection  then,  the  lens  L  will  bring  it  to  focus,  not  at  f1?  but 
at  the  new  point  f2,  such  that  the  straight  line  I2f2  passes 
through  the  center  of  L. 

The  eye  would  see  an  image  of  the  slit  at  fx  if  the  mirror 
were  at  rest,  at  f2  if  it  were  rotating.  In  the  latter  case,  the 
light  would  not  be  really  steady,  but  consist  of  a  series  of 
flashes;  but,  since  the  flashes  come  much  more  rapidly  than 
20  per  second,  it  would  to  all  appearance  be  steady.  If  the 
mirror  were  started  from  rest  and  gradually  picked  up  speed,  the 
image  would  first  appear  at  ft  and  gradually  move  away,  but 
would  seem  perfectly  still  as  long  as  the  speed  of  the  mirror 
were  steady. 

Evidently,  the  distance  fxf2  depends  upon  the  velocity  of 
light,  together  with  the  distance  between  the  two  mirrors  Mt 
and  M2,  the  speed  of  rotation  of  the  mirror  M1?  and  the  focal 
length  F;  consequently  we  should  be  able  to  get  the  velocity 
of  light  if  these  other  quantities  are  known.  The  small  dis- 
tance fjf,  is  measured  with  a  micrometer  (see  section  37),  the 
large  distance  MtM2  by  steel  tapes,  or  by  surveyors'  methods, 
and  the  speed  of  rotation  of  the  mirror  by  a  special  revolution- 
counter  and.  stop-watch,  or  some  equivalent  mechanical  device. 
The  focal  length  F  is  supposed  to  be  known,  or  it  can  be  found 
by  methods  to  be  described  later. 

In  order  to  derive  the  formula  for  finding  the  velocity  of 
light,  c,  we  shall  let  R  =  the  distance  IV^IVLj,  d  —  the  distance 


THE  ROTATING  MIRROR  METHOD  13 

f  tf2,  t  =  the  time  required  for  light  to  travel  the  distance  2R 
from  Mj  to  M2  and  back,  which  is  also  the  time  required  for 
the  mirror  to  turn  through  the  angle  a,  and  n  —  the  number 
of  revolutions  per  second  of  the  turning  mirror.  Then  the 
velocity  of  light  is 

2R 

c=  —  (1) 

L 

and  the  angular  velocity  of  the  mirror,  in  radians  per  second,  is 

2,rn  =  -  (2) 

t 

Since,  by  the  laws  of  reflection,  the  angle  A2MaM2  =  A^IJ.,, 

and  AJNIjM.,  =  A.MJ,,  therefore  LM^  =  2a.  In  the  actual 
experiment,  the  lens  L  is  placed  very  close  to  M1?  not  more 
than  a  few  feet  away,  while  the  distance  I^Ij  —  I2M,.  =  R  is 
quite  large,  say  several  hundred  meters.  Therefore,  very 
nearly,  the  angle  IXI,  =  IMJ.^  =  2a,  and  the  opposite  angle 
fxLf2  is  also  approximately  equal  to  2a.  With  Ix  and  I2  so 
far  from  the  lens,  the  images  f  1  and  f2  come  practically  at  the 
focal  distance  from  the  lens,  that  is  fxL  =  f,L  =  F.  There- 
fore. in  radian  measure, 


(3) 

Now,  from  equations  (1),  (2),  and  (3)  we  can  eliminate 
t  and  a  and  we  get 

c  =  87rnRF/d 

from  which  the  value  of  c  can  be  computed,  as  soon  as  n,  R, 
F,  and  d  are  measured. 

Professor  A.  A.  Michelson,  of  the  University  of  Chicago, 
has  made  a  number  of  improvements  in  the  details  of  Fou- 
cault's  method,  but  has  not  altered  the  principles  involved. 

.Measurements  of  the  velocity  of  light,  obtained  by  ex- 
perimental methods,  vary  from.  298,000  to  301,382  kilometers 
per  second.  It  is  usually  regarded  as  sufficiently  accurate  for 
all  purposes  to  take  the  round  figure  300,000,  or,  when  ex- 
pressed in  centimeters  per  second,  30,000,000,000  =  3  X  1010. 


14  LIGHT 

Problems. 

1.  The  star  Sirius  is  5  X  1013  miles  away  from  us.     How 
many  years  are  required  for  its  light  to  reach  us? 

2.  Calculate   the   time   required  for  light  to  travel   four 
miles  and  return  after  being  reflected  by  a  mirror. 

3.  What    is    the    maximum    angle    by    which,    owing    to 
aberration,    a   star   may   seem   to   be  displaced   from  its   true 
position  ? 

4.  Derive  the  formula  applying  to  Fizeau's  toothed- wheel 
method  for  finding  the  velocity  of  light. 

5.  Referring  to  figure  1,  at  what)  positions  of  the  earth 
does  the  observed  period  of  the  satellite  eclipses  seem  longest 
and  shortest  respectively? 

6.  It  frequently  happensi  that   the  moon  passes  between 
the  earth  and  a  star   (occultation  of  the  star).    What  would 
be  the  effect  upon  this  phenomenon  if  red  light  travelled  faster 
than  blue,  in  the  space  between  moon  and  earth? 

7.  The  "parallax"  of  a  star  is  the  angle  which  the  radius 
of  the  earth's  orbit.   92.8  X  106  miles,  subtends  as  seen  from 
the  star.    A  "parsec"  is  the  distance  of  a  star  whose  parallax 
is  one  second  of  arc.     Find  its  value  in  miles,  and  in  "light- 
years,"  the  distance  light  travels  in  a  year. 


CHAPTER  II. 


7.  Refraction  through  a  prism. — 8.  Newton's  conception  of  color. — 
9.  Impure  colors. — 10.  Color  due  to  absorption. — 11.  Color  due  to  other 
causes. — 12.  Black  and  white. — 13.  Complementary  colors  and  color 
mixture. — 14.  The  eye. — 15.  Color  vision  theories. 

7.  Refraction  through  a  prism. — Of  the  earlier  physicists, 
the  one  who  made  greatest  progress  in  the  study  of  light  was 
Sir  Isaac  Newton.  It  must  be  admitted  that  he  was  led  to 
believe  in  certain  hypotheses  which  have  since  been  discarded, 
but  in  spite  of  that  fact  he  accumulated,  by  experimental 
methods,  a  large  amount  of  needed  definite  information;  and 
his  philosophical  discussion  helped  greatly  in  the  development 
of  the  theory  that  later  supplanted  his  own  faulty  one. 

Newton  was  the  first  to  get 
a  clear  idea  of  color,  which  he      | 
attained    through    a    study    of      0 
glass  prisms.     Everyone  knows      w 
that  a  prism  of  any  transparent 
substance  not  only  bends  rays 
of  light,  but  also  makes  a  beam 
of  white  light  to  show  color  on 
the  edges.   Thus,  let  W  in  figure 
8  represent  a  window,  through 
which  white  light  enters  a  room, 
passing  through  the  prism  and 
entering   the    eye    placed    at    E 

general  way  the  course  of  the  rays.  Because  we  judge  the 
position  of  an  object  by  the  direction  of  the  rays  as  they 
enter  our  eyes,  the  window  appears  to  be  displaced  from  its 
true  position  to  some  such  place  as  W.  But,  more  than  this, 
the  window  appears  white  only  in  the  middle.  That  edge  of 
it  which,  as  seen  through  the  prism,  is  nearest  to  its  proper 
position,  is  red,  the  other  edge  violet.  Newton  saw  that  this 
experiment  indicates  white  light  to  be  a  composite  of  many 
colors,  the  color  effect  at  the  edges  being  a  result  of  some 
property  of  the  prism  which  causes  it  to  bend,  or  refract,  some 
of  these  component  colors  more  than  others;  for  instance,  the 

(15) 


r 
\v 


,r 


The 


Figure   8 

arrows    indicate    in    a 
we    judge 


16  LIGHT 

violet  more  than  the  red,  and  other  colors  to  an  intermediate 
degree.  According  to  this  hypothesis,  the  eye  would  see  a  red 
image  of  the  window,  as  indicated  by  the  rectangle  rrrr  in 
figure  9,  or  as  shown  in  the  plan  of  figure  8  by  rr;  while, 
slightly  displaced  from  it,  would  be  seen  a 
violet  image  (vvw  in  figure  9,  rv  in  figure  8). 
Any  other  color,  such  as  green,  would  also 
form  an  image  of  the  window,  displaced  less 
than  the  violet  but  more  than  the  red.  (Note 


that,  owing  to  a  certain  distorting  action  of 
the  prism,  which  we  shall  not  here  attempt 
to  explain,  the  vertical  edges  of  the  images  appear  not 
straight,  but  curved,  as  shown  by  the  dotted  lines  of  figure 
9).  Now,  if  we  bear  in  mind  that  there  exist  in  the  white 
light,  not  three  colors,  nor  only  seven,  but  an  infinite  number 
of  gradations  shading  into  one  another,  each  of  which  produces 
its  own  image  of  the  window,  it  is  easy  to  see  that  all  of  them 
will  overlap  in  the  middle,  so  that  this  part  will  be  white,  just 
like  the  light  as  it  enters  the  window.  But  on  passing  from 
the  middle  toward  one  edge,  we  find  first  the  violet  missing, 
then  the  colors  nearest  to  violet  (blue-violet,  blue,  etc.),  until 
finally,  at  the  extreme  edge,  only  the  red  is  present.  On  the 
other  hand,  passing  from  the  middle  toward  the  other  edge, 
first  the  red  is  missing,  then  the  intermediate  colors,  and  at 
the  extreme  edge  only  the  violet  remains.  It  is  clear  that  only 
the  extreme  colors,  red  and  violet,  are  seen  pure,  that  is,  un- 
mixed with  other  colors,  because  all  the  intermediate  ones  over- 
lap. But  it  is  also  evident  that  the  overlapping  would  be  very 
much  reduced  if,  instead  of  a  wide  window,  a  very  narrow 
slit  were  used  for  the  admission  of  the  light.  Since  it  is  im- 
possible to  use  an  infinitely  narrow  slit,  there  will  still  be  a 
small  amount  of  overlapping  of  the  images  produced  by  shades 
of  color  very  close  to  one  another,  but  none  at  all  in  the  case 
of  distinctly  different  colors.  This  can  be  understood  clearly 
if  the  reader  will  imagine  each  of  the  rectangles  of  figure  9, 
rrrr,  gggg,  vvvv,  etc.,  to  be  made  much  narrower,  without 
changing  the  distance  between  their  centers. 

Any  person  possessing  a  prism   can  try  this  experiment 
for  himself,  by  allowing  light  to  stream  through  the  crack  in 


NEWTON'S  COLOR  EXPERIMENTS 


17 


a  door  left  slightly  ajar,  and  viewing  the  crack  through  the 
prism  held  before  the  eye  as  in  figure  8,  with  the  refracting 
edge  vertical.  A  band  of  color  will  be  seen,  shading  from  violet 
at  one  edge,  through  blue,  green,  yellow,  and  orange,  to  red, 
at  the  other.  The  crack  in  the  door  acts  as  a  slit,  and  if  this 
be  narrow  enough  very  little  overlapping  will  occur  and  white 
will  nowhere  be  seen. 

Newton's  procedure  was  really  somewhat  different  from 
the  experiment  outlined  above.  He  allowed  a  beam  of  light 
direct  from  the  sun  to  pass  through  a  small  hole  O  in  a  shut- 
ter (figure  10)  and  then  through  a  prism  P,  which  deflected 

it  toward  the  white 

screen  S.  He  could 

have  placed  his  eye 

at    the    point    E, 

and  by  looking 

into    the    prism, 

seen  the  colored 

band  in  the  appar- 

e  n  t   position  r'v', 

since  the  red  light 

would  then  have 

entered    his    eve    as 


V'' 


Figure    10 

if  it  came  from  r',  the  violet  as  if  it 
came  from  v',  and  the  intermediate  colors  as  if  they  origi- 
nated at  points  intermediate  between  r'  and  v'.  Instead  of 
doing  this,  he  allowed  the  light  to  proceed  to  the  white  screen, 
forming  a  red  spot  at  r,  a  violet  spot  at  v,  etc.  Since;  the 
pencil  of  light  coming  from  the  sun  through  a  small  hole  is 
rather  narrow,  including  an  angle  of  only  about  one-half  de- 
gree, there  was  not  much  overlapping  of  the  colors,  and  the 
whole  colored  band  showed  the  intermediate,  as  well  as  the 
extreme  colors,  fairly  pure.  Newton  called  this  band  of  color 
a  spectrum,  and  in  technical  language  it  is  further  defined  as 
a  real  spectrum  because  the  light  actually  passes  through  it, 
or  at  least  to  it,  as  distinguished  from  the  so-called  virtual 
spectrum,  seen  in  the  apparent  position  rV  when  one  looks 
into  the  prism.  The  light  does  not  actually  pass  through  rV, 
but  merely  enters  the  eye  as  if  it  came  from  there.  There  is 
much  less  overlapping  of  colors  in  the  virtual  than  in  the  real 
spectrum, -in  Newton's  experiment,  that  is,  the  former  is  more 


18  LIGHT 

pure.  The  overlapping  in  the  virtual  spectrum  can  be  almost 
entirely  eliminated  by  making  the  hole  through  which  the  light 
is  admitted  very  small. 

The  width  of  the  spectrum  can  be  increased  by  replacing 
the  round  hole  with  a  narrow  slit,  and  the  overlapping  of 
colors  in  the  real  spectrum  can  be  much  reduced  by  the  inser- 
tion, either  before  or  behind  the  prism,  of  a  lens  of  suitable 
focal  length,  so  placed,  that  each  of  the  colors  is  brought  to  a 
focus  on  the  screen.  Under  these  circumstances,  it  is  possible 
to  regard  the  real  spectrum  as  made  up  of  an  infinite  number 
of  images  of  the  slit,  side  by  side,  the  color  of  each  image  being 
slightly,  though  imperceptibly,  different  from  that  of  the 
image  next  to  it.  If  the  light  coming  through  the  slit  contains 
(ivery  conceivable  gradation  of  color,  as  is  the  case  with  light 
coming  from  an  ordinary  electric  lamp,  there  will  be  no  gaps 
in  the  spectrum.  In  the  case  of  sunlight  there  are  certain 
missing  shades,  and  these  defects  are  made  evident,  if  the  slit 
is  very  narrow  and  the  focussing  very  good,  by  certain  gaps, 
or  "black  lines"  across  the  spectrum,  in  the  positions  of  those 
images  of  the  slit  which  would  be  supplied  by  the  missing 
colors  if  they  were  only  present,  (see  section  56) 

8.  Newton's  conception  of  color. — Newton's  conception  of 
the  formation  of  the  spectrum,  then,  was  that  the  differ- 
ent colors  are  already 
present  in  the  white 
light,  and  the  prism 
serves  only  as  a  separa- 
tor. This  view  was  direct- 
ly opposed  to  that  held 
by  some  others,  that  the 
prism  in  some  way  modi- 
fies the  light  so  as  to 
change  it  from  white  to 
Figure  11  colored.  According  to 

Newton's  ideas,  if  a  second  prism  be  introduced  behind  the 
first  one,  but  with  its  refracting  edge  turned  in  the  opposite 
direction,  as  in  figure  11,  this  should  reunite  the  colors  into  a 
white  beam  again.  A  trial  shows  that  this  actually  occurs, 
provided  the  second  prism  is  of  the  same  kind  of  glass,  and 
has  the  same  angle.  Another  test  of  Newton's  theory  is  this: 


IMPUBE   COLORS  19 

If  we  could  pass  through  a  prism  only  light  of  a  single  spectral 

tint,  for  instance  deep  red,  or  a  definite  shade  of  any  other 

color,  the  prism  should  simply  bend  it,  and  not  separate  it 

into  more  colors.    This  can  be  tried  by  the  arrangement  shown 

in  figure  12.    The 

white  screen,  S  of 

figure  10,  is  re- 

placed  by  a  screen 

in  which   there   is 

a  narrow  slit, 

through  which  any 

single    portion    o  f 

the  spectrum  can 

be  passed  to  a  sec-  Figure  12 

ond  prism.  It  is  found  that  the  second  prism  does  bend  the 
light,  but  does  not  spread  it  out  into  any  more  colors;  that  is, 
if  only  red  light  enters  it,  only  red  light  leaves  it,  and  that 
without  being  spread  out  to  any  appreciable  extent.  For  these 
experimental  reasons,  we  shall  accept  Newton's  view  of  color 
as  being  the  correct  one.* 

9.  Impure  colors- — There  is  one  fact  about  the  spectrum 
which  can  hardly  fail  to  strike  one  who  examines  it  closely, 
viz.,  that  we  fail  to  find  in  it  certain  colors  which  are  more  or 
less  common  in  nature.  For  instance,  there  is  no  purple,  and 
though  there  are  several  shades  of  red,  there  is  none  that  could 
be  called  pink.  These  two  cases  are  good  illustrations  of  the 
general  fact  that  most  of  the  colors  of  nature  are  not  ''pure 
colors, "  in  the  sense  of  spectral  colors  that  cannot  be  further 
separated  by  a  prism,  but  are  mixtures  of  two  or  more  of  these 
latter.  Purple,  for  example,  is  a  mixture  of  red  with  blue  or 
violet  or  both.  This  fact  can  be  shown  by  placing  two  small 

*Certain  modern  investigations  show  that  the  contention  of  New- 
ton's opponents,  that  a  prism  actually  manufactures  the  different 
colors  from  white  light  instead  of  merely  separating  out  constituents 
that  are  already  present,  is  in  a  certain  sense  true.  This  is  embodied 
in  what  is  called  the  pulse-theory  of  white  light.  But,  after  all  is  said, 
this  differs  from  Newton's  theory  only  in  the  point  of  view,  and  tht 
latter  not  only  explains  all  the  phenomena  in  a  satisfactory  manner, 
but  is  much  easier  to  deal  with.  Therefore  we  may  accept  it  as  true 
in  the  pragmatic  sense. 


20  LIGHT 

mirrors  in  the  spectrum,  one  in  the  deep  red  part,  the  other  in 
the  blue  or  violet  part,  turning  them  so  that  each  reflects  to 
the  same  spot  on  a  white  screen.  This  spot,  illuminated  by 
both  red  and,  blue  or  violet  light,  appears  purple.  Purple, 
then  is  a  sensation  produced  when  both  red  and  blue  or  violet 
light  fall  upon  the  retina  of  the  eye. 

We  can  get  a  pink  spot  by  illuminating  a  white  screen 
simultaneously  with  white  light  and  red  light.  Consequently, 
pink  is  produced  by  a  combination  of  all  the  spectral  colors, 
with  a  considerable  excess  of  red.  Similarly,  pale  blue  is  a 
combination  of  all  the  colors  with  an  excess  of  blue,  pale  green 
a  similar  combination  with  excess  of  green,  etc.  In  technical 
language,  any  such  combination  of  one  or  more  definite  spec- 
tral colors  with  white  is  said  to  be  unsaturated.  Thus,  pink 
is  unsaturated  red,  pale  blue  is  unsaturated  blue,  etc. 

That  the  pigments  used  ordinarily  in  painting  are  very 
impure  colors,  can  be  demonstrated  by  a  very  simple  experi- 
ment. Take  a  strip  of  paper  about  %2  inc^  wide,  and  color  it 
in  quarter-inch  lengths  with  the  following  artists'  pigments, 
each  pair  of  colors  being  separated  by  a  short  length  of 
white :  alizarin  crimson ,  alizarin  crimson  mixed  with  gam- 
boge, gamboge  alone,  gamboge  mixed  with  prussian  blue,  new 
blue,  and  alizarin  crimson  mixed  with  new  blue.  The  strip 
will  then  appear  as  a  very  narrow  ribbon  showing  the  follow- 
ing succession  of  colors :  red,  white,  orange,  white,  yellow, 
white,  green,  white,  blue,  white,  violet.  Now  lay  it  against 
a  dull  black  background,  such  as  a  piece  of  black  felt,  illumi- 
nate it  with  sunlight,  and  look  at  it  through  a  prism  whose 
edges  are  parallel  to  the  strip.  Each  of  the  white  portions 
forms  a  complete  spectrum,  with  which  the  spectra  of  the 
painted  portions  can  be  compared.  It  will  be  noted  that  each 
of  them  shows  through  the  prism  not  only  the  color  which  it 
appears  to  have  when  viewed  directly,  but  certain  other  parts 
of  the  spectrum.  Not  one  of  them  is  a  pure  color,  but  at  best 
each  shows  only  a  strong  excess  of  the  color  it  is  meant  to 
have.  The  most  nearly  pure  of  all  is  the  red  part,  but  even 
it  shows  quite  a  little  green,  with  traces  of  the  other  colors. 

A  narrow  blade  of  grass,  when  observed  through  a  prism 
in  such  a  manner,  shows,  beside  strong  green,  a  great  deal  of 
red  and  yellow,  and  even  some  blue  and  violet. 


COLOR  DUE  TO  ABSORPTION  21 

10.  Color  due  to  absorption. — Naturally  we  are  led  to 
enquire:  why  is  a  blade  of  grass  green,  or  the  petal  of  a  rose 
red?  'Since  grass  is  not  itself  luminous,  but  is  seen  only  be- 
cause it  reflects  diffusely  the  sunlight  that  falls  upon  it,  its 
green  color,  or  rather  its  very  mixed  color  with  green  predomi- 
nating, must  arise  from  the  fact  that  some  of  the  chemical 
substances  in  the  grass  have  the  property  of  "absorbing,"  to 
a  greater  or  less  extent,  certain  colors,  or  as  we  often  express 
it,  certain  parts  of  the  spectrum.  For  instance,  these  con- 
stituents of  the  grass  either  do  not  absorb  green  at  all,  or  more 
likely  simply  absorb  it  less  than  they  absorb  red  and  yellow, 
and  still  less  than  they  absorb  blue  and  violet.  But  absorption, 
in  the  proper  sense  of  the  word,  can  occur  only  while  the 
light  is  actually  passing  through  a  material,  not  in  the  mere 
act  of  reflection  at  the  surface.  Consequently,  it  must  be  that 
the  light  penetrates  the  surface  to  some  appreciable  depth,  that 
is,  the  substance  of  the  grass  is  to  some  extent  transparent. 
That  this  is  true,  can  be  readily  proved  by  holding  a  blade  of 
grass  between  the  eye  and  a  bright  source  of  light.  A  con- 
siderable fraction  of  the  incident  light  passes  entirely  through 
the  grass  to  the  eye.  Indeed,  it  may  truly  be  said  that  any 
material  is  to  some  extent  transparent,  and  if  made  into  a 
thin  enough  sheet  will  allow  an  appreciable  amount  of  light  to 
pass  through  it.  But  the  farther  the  light  is  caused  to  pass 
through  a  material  the  more  of  its  energy  is  absorbed. 

Evidently,  what  happens  in  the  case  of  the  blade  of  grass 
is  something  like  this :  Of  the  white  light  that  strikes  the  sur- 
face, part  is  reflected  without  penetrating,  as  would  be  the  case 
with  glass  or  water;  and  this  part,  if  it  could  be  seen  alone, 
would  be  white,  like  the  incident  light.  The  rest  of  the  incident 
light  penetrates  the  surface;  but  since  the  material  is  not 
completely  transparent,  but  only  what  we  call  translucent,  the 
rays  do  not  pass  straight  through  to  the  back  surface,  but  are 
diffused,  or  scattered,  within;  the  material.  In  this  way  part 
of  the  light  eventually  gets  back  into  the  air  through  the  front 
surface,  and  we  call  this  part  "diffusely  reflected"  light, 
although  it  has  been  within  the  body  of  the  material.  Part 
also  gets  out  through  the  back  surface,  and  we  call  this 
"transmitted"  light.  Both  the  diffusely  reflected,  and  the 
transmitted,  light  during  its  passage  through  the  material, 


22  LIGHT 

suffers  losses  by  absorption  in  the  chemical  substances  of  the 
leaf.  Part  of  the  green  is  absorbed,  more  of  the  yellow,  orange, 
and  red,  and  most  of  the  blue  and  violet.  In  this  way,  both 
the  reflected  and  the  transmitted  light  become  colored,  and 
indeed  both  parts  show  about  the  same  color. 

Most  natural  objects  owe  their  colors  to  the  same  cause. 
Part  of  the  light  penetrates  the  surface,  and  part  of  this 
emerges  again,  after  suffering  absorption.  This  explanation  is 
satisfactory  so  far  as  it  goes,  but  it  must  not  be  forgotten  that 
we  do  not  know  why,  for  instance,  the  leaf-substance  absorbs 
more  red  than  green,  while  just  the  reverse  is  true  of  the 
coloring-material  known  as  alizarin-crimson.  This  question 
cannot  be  answered  without  a  far  greater  knowledge  of  atomic 
structure  than  we  now  have. 

11.  Color  due  to  other  causes. — There  are  some  objects 
whose  color  is  produced  in  a  different  way.  The  yellow  color 
of  gold,  as  an  example,  is  mostly  due  to  the  fact  that  a  certain 
shade  of  yellow  light  seems  unable  to  enter  the  gold  at  all.  It 
is  completely  reflected  at  the  surface.  Consequently,  a  very 
thin  sheet  of  gold-leaf  transmits  light  which  is  completely 
lacking  in  this  color.  The  transmitted  light  is  dull  green,  while 
the  reflected  light  is  yellowish.  A  similar  phenomenon  occurs 
in  the  case  of  some  dyes,  which  in  concentrated  form  show 
quite  a  different  color  according  as  they  are  seen  by  transmitted 
or  by  reflected  light.  Common  red  ink  is  an  example;  it 
reflects  green  very  strongly  when  concentrated,  but  transmits 
red. 

The  brilliant  colors  of  rainbows  are  due,  not  at  all  to 
absorption,  but  to  a  separation  of  the  colors  something  like 
that  which  occurs  in  a  prism.  The  theory  of  rainbows  will  be 
taken  up  later,  (section  83). 

The  blue  of  the  sky  is  caused  by  a  sort  of  scattering  of 
the  light  by  the  particles  of  the  atmosphere,  similar  to  the 
scattering  produced  when  a  beam  of  light  is  sent  through  milky 
water.  If  there  were  no  atmosphere  the  sky  would  appear 
black,  and  we  would  receive  light  only  from  the  sun,  moon, 
and  stars  directly.  In  the  scattered  light  from  the  sky,  all 
colors  of  the  spectrum  are  represented,  with  blue  in  excess. 
In  order  to  explain  this  preponderance  of  blue,  we  must 
anticipate  to  some  extent  facts  that  properly  come  later  in 


BLACK  AND  WHITE  23 

these  pages.  It  will  be  shown  in  the  next  chapter  that  light 
consists  of  waves,  the  shortest  of  which  are  the  violet,  the 
longest  the  red,  the  intermediate  colors  having  waves  of  inter- 
mediate length.  Although  all  these  waves  are  very  short,  the 
length  of  even  the  shortest"  is  much  greater  than  any  of  the 
dimensions  of  a  molecule.  Since  the  molecules  are  so  small, 
they  are  much  more  efficient  in  reflecting,  or  scattering,  short 
waves  than  longer  ones,  just  as  small  pieces  of  wood  floating 
on  the  surface  of  water  will  reflect  short  ripples,  but  simply 
ride  on  the  very  long  waves.  Therefore  the  molecules  in  the 
air  reflect — scattering  in  all  directions — those  colors  that  lie 
near  the  violet  end  of  the  spectrum  to  a  considerably  greater 
degree  than  those  near  the  red  end,  thus  giving  a  bluish  color 
to  this  scattered  light.  That  it  appears  blue  rather  than  violet 
is  because  the  violet  is  at  best  very  weak. 

Of  course,  since  a  beam  of  direct  light  from  the  sun  is 
thus  robbed  of  a  greater  percentage  of  its  blue  and  violet  than 
of  its  red,  that  part  of  it  which  passes  on  through,  must  be 
abnormally  rich  in  red — relatively  speaking — and  therefore  / 
must  appear  more  reddish  in  color  than  it  was  when  it  emerged 
from  the  sun.  This  is  particularly  marked  when  the  light  has 
passed  through  a  long  distance  in  air  before  reaching  the  eye, 
as  is  the  case  near  sunrise  or  sunset.  That  the  light  at  sucix 
times  is  exceptionally  impoverished  in  blue  and  violet,  is  well 
known  to  every  photographer,  for  most  of  the  photographic 
action  of  light  is  produced  by  these  colors,  and  a  plate  must 
be  exposed  several  times  as  long  when  the  sun  is  low  in  the 
sky  as  at  midday.  However,  the  reddish  color  of  the  sun  when 
it  is  near  the  horizon  is  familiar  to  all. 

12.  Black  and  white. — A  black  object,  strictly  speaking, 
is  one  which  absorbs  completely  all  colors,  and  reflects  none. 
But  an  object  may  appear  black  simply  because  the  light 
which  illuminates  it  contains  no  constituent  save  those  which 
it  absorbs  completely.  For  example,  a  deep  red  rose  will  ap- 
pear black  when  placed  in  the  blue  or  violet  part  of  the  spec- 
trum, because  it  absorbs  these  colors  completely,  and  the  only 
color  which  it  can  reflect  freely,  red,  is  not  present. 

A  -ibhite  object  is  one  which  reflects  diffusely,  that  is  ii? 
all  directions,  all  colors  to  the  same  extent.  The  whiteness  of 
the  snow  is  an  interesting  case.  Snow  is  really  composed  of 


24  LIGHT 

numerous  little  ice  crystals,  and  ice  in  bulk  is  not  a  white 
body,  but  ai  transparent  one.  That  is,  a  large  chunk  of  ice 
allows  most  of  the  light  which  falls  upon  it  to  pass,  through, 
and  that  part  which  it  reflects  is  reflected,  not  diffusely,  but 
in  a  definite  direction.  But  when  we  have  a  great  mass  of  very 
small  ice  crystals,  arranged  in  an  irregular  manner,  although 
each  crystal  surface  reflects  in  a  particular  direction,  the  whole 
mass  reflects  about  as  much  in  one  direction  as  in  another. 
Moreover,  the  light  that  passes  through  the  crystals  on  top  of 
the  layer  of  snow  will  strike  other  surfaces  below,  which  again 
cause  reflection,  part  of  the  reflected  light  finding  its  way  out 
of  the  mass  again.  Thus  the  whole  mass  reflects  irregularly  a 
very  large  amount  of  the  light,  and,  since  no  color  is  absorbed 
by  the  material,  this  reflected  light  is  white  in  color.  There- 
fore, the  whiteness  of  snow  is  due  to  the  great  number  of  re- 
flecting surfaces,  irregularly  arranged.  The  same  effect  can 
be  produced  by  crushing  a  piece  of  glass  with  a  hammer.  The 
many  cracks  in  the  glass  cause  a  multitude  of  reflecting  sur- 
faces, arranged  irregularly,  and  the  mass  immediately  becomes 
white.  The  whiteness  of  clouds  is  also  due  to  numberless  little 
reflecting  surfaces,  the  surfaces  of  millions  of  small  water- 
drops. 

Of  course,  a  white  object  will  not  appear  white  unless  the 
light  which  falls  upon  it  contains  all  the  colors  of  the  spectrum, 
i.  e.,  is  white  light.  If  it  be  illuminated  by  red  light  it  will 
appear  red,  if  by  blue  light,  blue,  etc.  Thus,  the  white  screens 
of  figures  10  and  11  show  whatever  color  falls  upon  them,  and 
it  is  just  this  property  that  makes  a  white  screen  suitable  for 
such  experiments. 

13.  Complementary  colors  and  color  mixture. — Any  two 
colors  which  together  produce  the  sensation  of  white  are  called 
complementary  colors.  A  convenient  way  of  showing  these  is 
illustrated  in  figure  13.  White  light  passes  through  the  slit 
S  to  the  lens  L,  which  makes  the  rays  parallel.  It  then  passes 
through  the  prism  P  and  the  second  lens  L2,  which'  focusses 
the  spectrum  in  the  plane  vr.  Instead  of  having  a  screen  at 
this  place,  another  lens,  L3,  is  placed  just  behind  it.  The  two 
lenses  L2  and  L3  together  form,  an  image  of  the  face  of  the 
prism  on  a  properly  placed  white  screen  A.  Since  light  of  all 
colors  comes  through  the  whole  face  of  the  prism,  and  all  the 


COLOR  1VIIXTURE  25 

light  passes  through  the  two  lenses,  this  image  will  be  uniform- 
ly white.  Now,  if  an  opaque  obstacle  be  placed  just  in  front 
of  L,.  i.  e.,  just  in  the  plane  in  which,  the  spectrum  is  formed, 
the  obstacle  being  of  sufficient  width  to  cut  off  a  certain  spec- 
tral region,  say  the  green,  then  only  the  remaining  colors  will 
reach  the  screen  A,  and  the  image  of  the  prism-face  will 
therefore  show  the  color  complementary  to  the  color  cut  out. 


Figure   13 

By  this  means,  we  find  that  the  color  complementary  to  the 
average  spectral  green  is  a  peculiar  shade  of  red,  that  com- 
plementary to  spectral  blue  a  sort  of  golden  yellow,  etc. 
Generally,  one  or  both  of  a  pair  of  complementary  colors  are 
impure  in  the  spectral  sense. 

It  is  found  that  a  combination  of  the  red,  the  green,  and  / 
the-Jalue  of  the  spectrum,  in  suitable  proportions,  will  produce 
the  sensation  of  white,  without  the  presence  of  the  other  colors, 
such  as  violet,  orange,  and  yellow.  This  can  be  tried  with  the 
arrangement  of  figure  13  by  placing  before  the  lens  L3  a  card 
with  holes  cut  through  it  so  as  to  let  pass  only  some  of  these 
three  colors.  The  best  exact  location  of  the  holes,  and  their 
proper  sizes,  can  be  found  only  by  trial.  Furthermore,  by 
suitably  altering  the  relative  intensities  of  these  three  colors, 
as  by  stopping  down  one  or  two  of  the  holes,  any  other  color, 
either  a  pure  spectral  hue,  or  such  a  color  as  purple,  can  be 
closely  imitated. 

The  mixing  of  pigments  shows  some  results  which  at  first 
sight  are  very  surprising.  For  instance,  since  a  combination 
of  all  the  different  colors,  of  proper  proportions,  produces 
white,  we  should  naturally  expect  that  when  many  different 
paints  are  mixed,  the  mixture  would  tend  to  become  white. 


26  LIGHT 

On  the  contrary,  it  tends  to  become  black;  and  in  general, 
the  more  different  pigments  are  put  into  a  mixture,  the  darker 
it  becomes.  The  reason  is  that  by  mixing  we  combine  the 
absorbing  powers  rather  than  the  reflecting  powers  of  the 
constituents.  We  have  seen  that  the  common  pigments,  alizarin 
crimson,  gamboge,  and  new  blue,  have  each  a  distinctive  ab- 
sorption, and  in  a  mixture  of  the  three  any  part  of  the  spec- 
trum would  be  strongly  absorbed  by  one  or  another,  so  that 
little  if  any  of  the  incident  light  would  escape  absorption.  An 
example  less  extreme  than  this  is  seen  in  the  mixture  of  the 
crimson  and  the  blue  to  produce  violet.  If  it  were  not  for  the 
fact  that  certain  parts  of  the  spectrum  escape  complete  absorp- 
tion in  either  of  these  pigments,  the  mixture  would  be  black 
instead  of  violet.  It  is,  as  a  matter  of  fact,  extremely  dark, 
much  darker  than  the  violet  seen  in  the  spectrum  from  direct 
sunlight,  relative  to  the  brightness  of  the  other  colors. 

It  is  quite  plain  then,  that  the  mixing  of  two  or  mx>re 
paints  produces  quite  a  different  result  from  throwing  simul- 
taneously upon  a  white  screen  lights  of  the  corresponding 
colors;  and  this  fact  seriously  limits  the  ability  of  artists  to 
produce  desired  effects  by  mixing  paints.  The  school  of 
painters  known  as  impressionists  introduced  a  new  method. 
Instead  of  mixing  their  paints,  they  lay  them  on  the  canvas 
in  little  blotches  side  by  side.  Thus,  where  a  painter  of  the 
older  schools  would  mix  crimson  and  yellow  to  paint  a  surface 
of  orange  color,  the  impressionist  covers  the  surface  with  dots 
of  crimson  and  dots  of  yellow  close  together  but  arranged  in 
irregular  order.  Such  a  painted  surface  looks  very  confusing 
when  viewed  at  close  range,  but  at  a  greater  distance  the 
blotches  of  red  and  yellow  seem  to  blend  together  to  produce 
the  effect  of  a  uniform  orange,  so  that  the  impressionist 
secures  in  this  way  the  same  effect  that  we  could  get  in  the 
laboratory  by  simultaneously  illuminating  a  white  surface  with 
red  and  yellow  light.  As  a  result,  paintings  by  impressionists 
are  usually  far  more  brilliant  than  those  of  the  old  masters, 
though  it  is  true  that  the  latter  have  a  sombre  richness  which 
is  itself  a  great  charm. 

14.  The  eye. — The  organ  of  vision,  the  eye,  is  an  optical 
instrument  more  analogous  to  the  photographic  camera  than  to 


THE  EYE.  27 

anything  else.  It  is  shown  diagrammatically  in  figure  14.  It 
consists  of  a  shell  roughly  spherical  in  shape,  of  which  the 
front  wall,  C.  the  cornea,  is  transparent.  Behind  this  is  the 
iris  I,  a  screen  or  diaphragm  con- 
taining a  circular  hole, — the  pupil, 
P, — whose  diameter  contracts  in 
brilliant  illumination  or  expands  in 
dim  light,  by  involuntary  muscular 
action.  The  lens  L  is  capable  of  a 
slight  forward  and  backward  mo- 
tion, like  that  of  a  camera  lens  in 
focussing,  but  most  of  the  focus- 
sing in  the  eye  is  accomplished  by  altering  the  radii  of 
the  lens  surfaces.  The  material  of  the  lens  is  of  course  some- 
what plastic,  and  in  structure  it  resembles  an  onion  in  being 
built  up  in  layers.  The  lens  forms  an  image  of  any  object 
looked  at  upon  the  retina  R,  which  is  spread  over  the  rear  and 
side  surfaces  of  the  shell.  The  space  between  the  lens  and  the 
retina  is  filled  with  a  jelly-like  material  called  the  vitreous 
humor.  The  material  between  lens  and  cornea  is  watery,  and 
is  called  the  aqueous  humor.  By  means  of  the  above  mentioned 
muscular  distortion  of  the  lens,  together  with  its  slight  axial 
movement,  any  object  toward  which  the  eye  is  directly  turned 
can  be  sharply  f ocussed  upon  the  retina,  provided  it  is  not 
closer  than  a  few  inches,  in  the  case  of  normal  eyes.  This  ad- 
justment of  focus  is  called  accommodation.  With  the  eye  re- 
laxed, very  distant  objects  should  be  in  sharp  focus.  In  view- 
ing a  very  small  object,  however,  it  is  advantageous  to  bring 
it  close,  so  that  its  image  upon  the  retina  may  be  larger;  but 
if  it  is  brought  closer  than  about  10  inches  the  muscular  strain 
of  accommodation  becomes  unpleasant  and  is  in  fact  harmful. 
Consequently,  about  10  inches  (25  cm.)  is  the  most  favorable 
distance,  with  people  of  normal  vision,  for  reading  or  for 
careful  scrutiny  of  small  objects.  This  is  known  as  the  "  dis- 
tance of  distinct  vision." 

A  myopic,  or  short-sighted,  eye  is  one  whose  focal  length 
in  the  relaxed  condition  is  abnormally  short,  so  that  it  cannot 
focus  sharply  upon  distant  objects.  Just  the  reverse  is  a 
hypermetropic  eye,  which  requires  a  certain  amount  of  accom- 


28  LIGHT 

modation  even  to  focus  upon  an  infinitely  distant  object. 
Presbyopia,  a  trouble  particularly  common  among  older  people, 
is  an  impairment  of  the  ability  to  accommodate,  due  to  a  pro- 
gressive stiffening  of  the  muscles  which  change  the  radii  of 
the  lens.  Astigmatism  is  caused  by  lack  of  axial  symmetry  in 
the  lens  or  the  cornea  or  both.  It  shows  itself  in  an  inability 
to  see  clearly,  with  the  same  accommodation,  lines  inclined  at 
different  degrees  with  the  vertical,  though  equally  distant. 

The  focus  upon  the  retina  is  sharp  only  for  a  limited 
region  near  the  axis  of  the  lens,  but  unsharp  vision  is  possible 
over  a  very  wide  angle  without  moving  the  eye. 

The  retina  is  the  sensitive  part  of  the  eye,  which  in  some 
manner  is  stimulated  by  light  falling  upon  it  so  that  the  mind 
experiences  the  sensation  of  vision.  It  is  a  network  of  delicate 
nerve-fibers  which  are  connected  with  the  brain  through  the 
optic  nerve  O. 

15.  Color  vision  theories. — The  student  should  understand 
that  the  actual  mechanism  of  vision,  the  connecting  link  be- 
tween the  light-stimulus  upon  the  retina  and  the  consciousness 
of  light  and  color,  is  a  thing  about  which  little  is  known.  Even 
if  we  knew  the  physical  and  chemical  processes  that  go  on  in 
a  iierve,  there  would  still  be  a  gap  or  hiatus  in  our  knowledge 
between  that  and  the  actual  sensation.  Consequently,  our 
notions  of  light  and  color  perception  do  not  extend  very  far, 
and  are  somewhat  uncertain  at  that.  On  the  face  of  things, 
it  seems  very  unlikely  that  there  is  a  separate  type  of  nerve 
for  every  gradation  of  color.  Such  an  experiment  as  the  pro- 
duction of  the  orange  sensation  by  mixing  red  and  yellow  light, 
or  the  production  of  any  other  spectral  sensation  by  a  mix- 
ture in  suitable  proportions  of  red,  green  and  blue,  suggests 
that  there  are  only  a  few  distinct  color  sensations,  perhaps 
three,  and  that  the  other  sensations  are  the  result  of  a  simul- 
taneous stimulation  of  these  few.  Experiments  with  color- 
blind people  also  support  this  view.  There  are  in  fact  two 
principal  theories  of  color-sensation,  the  Young-Helmholtz 
theory  and  the  Hering  theory.  The  former  alone  is  often 
given  in  physics  texts,  but  the  latter  is  favored  by  at  least  a 
great  many  experimental  psychologists,  and  the  matter  is  one 
of  psychology  more  than  of  physics. 


COLOR- VISION  THEORIES.  29 

According  to  the  Young-Helmholtz  theory,  the  retina  con- 
tains three  distinct  sets  of .  nerve-fibers,  each  giving  only  a 
single  sensation,  no  matter  what  particular  part  of  the  spec- 
trum corresponds  to  the  light  that  does  the  stimulating.  One 
set  gives  a  red  sensation,  the  second  a  green  sensation,  the 
third  a  violet  or  blue  sensation.  The  three  curves  of  figure 
15,  which  are  due  to  Koenig,  show  to  what  extent  each  of  these 


Dp.  R 


sensations  is  stimulated  by  light  from  different  parts  of  the 
spectrum.  In  order  to  clearly  understand  these  curves,  con- 
sider how  the  spectrum  would  appear  to  a  person  whose  eyes 
were  provided  with  the  red-sensitive  nerves,  but  not  with  the 
other  two  sets.  He  would  be  able  to  see  the  spectrum  through- 
out its  entire  extent,  with  the  possible  exception  of  the  ex- 
treme violet  end,  but  all  of  it  would  appear  of  the  same  color, 
red.  The  only  differences  between  different  parts  would  be 
differences  of  brightness,  as  indicated  by  the  varying  ordinates 
of  the  red  curve;  In  fact,  his  retina  would  act  in  a  way  quite 
analogous  to  the  action  of  a  photographic  plate,  which  responds 
to  the  influence  of  light  of  many  different  colors,  but  with  a 
response  which  is  the  same  in  kind  for  all,  differing  only  in 
degree. 

Now  consider  an  eye  with  all  three  sets  of  nerve-fibers, 
and  suppose  the  retina  to  be  stimulated  by  light  from  the 
yellow-green  portion  of  the  spectrum.  A  comparison  of  figure 
15  shows  that  this  light  stimulates  all  three  of  the  sensations, 
the  green  sensation  most  strongly,  the  red  sensation  to  a  lessi 
degree,  and  the  violet  sensation  least  of  all.  The  complex  of 
these  three  sensations  acting  together  is  what  we  are  accustomed 
to  call  the  yellow-green  sensation.  About  one  man  in  thirty 


30  LIGHT 

is  "red-color-blind/5  which,  on  the  Young-Helmholtz  theory, 
means  that  his  eyes  lack  the  nerve-fibers  which  give  the  sen- 
sation of  red. 

The  Hering  theory  is  quite  different.  Instead  of  three 
primary  sensations,  it  postulates  certain  contrasts,  caused  by 
chemical  changes,  under  the  influence  of  light,  in  three  hypo- 
thetical fluids  present  in  the  retina,  which  we  shall  designate 
as  A,  B,  and  C.  Fluid  A  undergoes  a  certain  decomposition 
when  any  sort  of  light,  irrespective  of  color,  falls  upon,  it,  but 
recombines,  or  recovers,  in  darkness.  It  reacts  upon  the  nerve 
fibers  differently  in  its  two  states,  causing  a  sensation  of  bright- 
ness in  the  one  case  and  of  darkness  in  the  other.  Fluid  B 
is  different.  It  undergoes  a  decomposition  under  the  action 
of  light  of  longer  wavelengths,  giving  a  red  sensation,  and  a 
recombination  under  the  action  of  light  of  shorter  wavelengths, 
giving  a  green  sensation,  being  entirely  neutral  for  light  of 
wavelength  corresponding  to  some  part  of  the  yellow.  Fluid 
C  acts  in  a  similar  way,  but  the  sensation  produced  by  longer 
wavelengths  is  yellow,  that  by  shorter  wavelengths  blue  or 
violet,  and  the  neutral  condition  would  be  for  wavelengths  in 
the  green.  Thus  we  have  three  contrasts,  bright  and  dark,  red 
and  green,  yellow  and  blue.  According  to  this  theory,  the 
usual  type  of  color-blindness  is  due  to  lack  of  the  B  fluid, 
resulting  in  an  inability  to  distinguish  reds  from  greens. 

Color-blindness  is  not  a  disease,  but  a  heritable  defect,  and 
though  a  handicap  it  is  not  a  thing  of  which  one  need  be 
particularly  ashamed.  Many  people  who  have  it  are  not  con- 
scious of  it.  Recent  biological  researches  have  shown  the  fol- 
lowing interesting  peculiarities  about  its  inheritance.  A 
color-blind  man  transmits  the  defect  neither  to  his  sons  nor 
to  his  daughters,  but  to  the  sons  of  his  daughters;  that  is,  it 
passes  from  the  male  of  the  first  generation  to  the  male  of  the 
third,  through  the  female  of  the  second,  but  without  showing 
actively  in  the  second  generation  at  all.  A  woman  is  never 
herself  color-blind  unless  she  inherits  it  both  from  her  father 
and  her  mother's  father.  Consequently,  cases  of  color-blind- 
ness among  women  are  very  rare. 


COLOR-VISION  THEORIES  31 


Problems. 

1.  Suppose  that  a  flower  whose  color  is  a  pure  blue  is 
passed  slowly  through  a  spectrum,  from  one  end  to  the  other. 
What  would  be  its  appearance  in  the  different  parts?     Suppose 
a  blade  of  grass  were  treated  in  the  same  manner? 

2.  A  story  by  Ambrose  Bierce,  entitled    "The    Damned 
Thing,"  has  for  its  subject  a  supposedly  invisible  animal.    The 
author    argues    that    such    a    thing    would  be  possible  if  the 
animal's  fur  reflected  only  ultraviolet  light.     What  would  be 
the  actual  appearance  of  such  an  animal  f 

3.  Explain  the  whiteness  of  soapsuds  and  other  froth. 

4.  Suppose  blue  glass  were  crushed  to  a  powder.     What 
would  be  the  effect  upon  its  color? 

5.  Explain  why  tobacco    smoke    appears    blue    against    a 
dark,   but  brown   against   a  bright,  background. 

6.  Why  is  it  that  colored  cloth  can  be  changed  by  dyeing 
to  a  darker,  but  not  to  a  brighter  color? 

7.  Birds,  animals,  and  fishes  usually  have  a  much  lighter 
color  on  the  lower  than  on  the  upper  sides  of  their  bodies.    Is 
this  fact  of  any  importance  in  the  economy  of  nature?   Explain. 


CHAPTER  III.  . 

16.  The  corpuscular  theory  of  light. — 17.  The  wave  theory. — 18. 
Bending  of  light  into  a  shadow.— 19.  Nature  of  the  ether. — 20.  Waves 
in  general.  Plane  waves. — 21.  Mathematical  formula  for  a  wave. — 22. 
Interference.  Fresnel's  mirrors. — 23.  Interference  in  white  light. 

16.  The  corpuscular  theory  of  light. — We  have  now  learned 
enough  about  some  of  the  general  properties  of  light  to  en- 
quire with  some  degree  of  intelligence  as  to  its  nature.  The 
two  theories  that  have  had  any  support  may  be  called  the  cor- 
puscular theory  and  the  wave  theory.  According  to  the  first, 
light  consists  of  very  small  weightless  material  particles;  ac- 
cording to  the  second,  it  consists  of  waves.  Either  theory 
strains  the  imagination  greatly. 

It  is  hard  to  think  of  material  corpuscles  flying  with 
enormous  speed  through  a  solid  substance  like  glass,  with  so 
little  hindrance  as  glass  seems  to  offer  to  the  passage  of  light, 
though  color  might  well  be  accounted  for  by  differences  in 
size,  in  shape,  or  in  some  other  characteristic  among  the  cor- 
puscles. It  is  also  extremely  difficult  to  explain  how,  when 
these  corpuscles  strike  such  a  substance  as  glass  or  water,  some 
of  them  should  be  reflected  while  others  pass  into  the  material, 
being  refracted  as  they  do  so.  It  is  true  that  one  might  sup- 
pose that  there  are  two  kinds  of  such  particles,  a  kind  that 
is  reflected  and  a  kind  that  is  refracted.  But  if  this  were  the 
case,  one  reflection  would  completely  separate  these  two  kinds, 
so  that  if  the  reflected  light  struck  another  such  surface  all  of 
it  would  be  reflected,  none  refracted;  while  if  the  refracted 
light  struck  another  surface,  all  would  be  refracted,  none  re- 
flected. Then  if  one  should  observe  in  a  plate  glass  window 
the  reflected  image  of  his  own  body,  this  image  would  become 
invisible  to  him  if  he  held  a  small  piece  of  glass  in  front  of  his 
eyes.  A  simple  trial  shows  that  this  is  not  true.  Further- 
more, the  light  that  got  in  through  the  first  surface  of  the 
plate  glass  could  not  be  reflected  at  all  by  the  second  surface. 
It  can  easily  be  proved  that  this  conclusion  also  is  false,  for 
if  one  stands  close  to  such  a  window  he  can  distinctly  see  two 
images  of  himself,  the  brighter  image  being  formed  by  re- 

(32) 


: 


THE  WAVE  THEORY  33 


; lection  at  the  first  surface,  the  fainter  one  at  the  second.  (In 
reality  there  is  a  large  number  of  such  images,  produced  by 
multiple  reflections,  but  all  except  the  first  two  are  very  faint). 
In  order  to  get  around  this  difficulty,  Newton,  the  chief  advo- 
cate of  the  corpuscular  theory,  suggested  that,  although  all 
the  corpuscles  are  fundamentally  alike  so  far  as  reflection  and 
refraction  are  concerned,  each  one  is  at  times  in  a  state  suitable 
for  reflection,  at  other  times  in  a  state  suitable  for  refraction; 
so  that  whenever  the  light  strikes  a  reflecting  surface  there 
will  be  a  certain  proportion  of  them  ready  for  reflection,  even 
though  some  had  already  been  reflected  before.  But  such  an 
hypothesis,  though  not  absolutely  absurd,  seems  clumsy  and 
improbable,  and  Newton  himself  was  far  from  being  satisfied 
with  it. 

17.  The  wave  theory. — In  the  wave  theory,  there  is  no 
difficulty  in  explaining  reflection  and  refraction.  Indeed  it  is 
characteristic  of  all  kinds  of  wave  motion  that,  whenever  a 
wave  strikes  a  surface  separating  two  media  in  which  the 
velocity  of  wave  propagation  is  different  (such  as  the  surface 
between  air  and  glass)  part  of  the  energy  enters  the  second 
medium  as  a  refracted  wave,  while  part  is  sent  back  into  the 
first  as  a  reflected;  wave.  Neither  is  it  at  all  hard  to  think 
of  waves  passing  through  glass  and  other  transparent  media 
with  high  velocity  and  little  resistance,  for  we  know  that 
mechanical  waves,  such  as  sound  waves,  do  traverse  such 
bodies  very  easily.  Color,  also,  may  be  accounted  for  very 
simply  on  the  wave  theory,  by  the  supposition  that  differences 
in  color  correspond  to  differences  in  the  length  of  the  waves, 
just  as  we  know  that  differences  in  the  pitch  of  musical  notes 
correspond  to  differences  in  the  lengths  of  the  sound  waves. 

But  it  is  difficult  to  understand  how  waves  can  pass,  as 
we  know  that  light  does,  through  perfectly  empty  space,  for 
the  use  of  the  names  /'wave"  implies  the  existence  of  some 
medium  in  which  the  waves  exist.  On  account  of  this  difficulty 
with  the  wave  theory,  physicists  have  been  led  to  assume  the 
existence  of  a  medium  filling  all  space,  even  a  so-called  vacuum, 
to  which  the  name  ether  has  been  given.  The  necessity  for 
this  assumption  is  to  this  day  a  very  serious  load  on  the 
shoulders  of  the  wave  theory  of  light,  though  it  becomes  less 


34 


LIGHT 


objectionable  when  we  find  that  there  are  other  phenomena; 
such  as  electric  and  magnetic  attractions  and  repulsions,  which 
also  operate  through  a  vacuum,  and  also  seem  to  indicate  the 
existence  of  some  all-pervading  medium. 

.  Newton's  chief  objection  to  the  wave  theory,  however, 
was  not  the  necessity  for  an  ether,  but  the  fact  that  light 
apparently  travels  in  straight  lines,  while  other  waves,  such 
as  sound  waves,  or  waves  on  the  surface  of  water,  will  bend 
freely  round  an  obstacle  placed  in  their  path.  In  this  con- 
nection, it  should  be  noted  that  the  degree  to  which  waves 
bend  round  an  obstacle,  thus  departing  from  the  straight-line 
path,  depends  to  a  great  extent  upon  the  length  of  the  waves. 
Long  waves  do  so  much  more  freely  than  short  waves.  There- 
fore, if  it  can  be  shown  that  light  really  does  to  some  slight 
extent  bend  round  a  corner,  this  objection  will  be  overcome, 
and  at  the  same  time  evidence  will  be  acquired  that  if  light 
does  consist  of  waves  these  waves  are  very  short. 

18.  Bending  of  light  into  a  shadow. — In  fact,  the  bending 
of  light  about  an  obstacle  can  be  shown,  by  a  very  simple 
experiment  illustrated  in  figure  16.  0  represents  any  opaque 

obstacle  with  a  straight  edge 
perpendicular  to  the  plane 
of  the  paper.  The  source  of 
light,  S,  must  be  of  very 
small  dimensions ;  otherwise 
the  bending  of  the  light  into 
the  shadow,  which  the  ex- 
periment is  designed  to  show, 
will  be  masked  by  the  pe- 

numbral  effect  always  shown  when  a  shadow  is  cast  by 
a  fairly  large  source  of  light.  It  should  be  either  a  very 
fine  hole,  or  better  still  a  very  narrow  slit,  illuminated 
by  an  arc  light  or  other  brilliant  illuminant.  A  is  a  white 
screen,  to  receive  the  shadow.  Let  a  straight  line  SQ  be  drawn 
from  the  slit  through  the  edge  of  the  obstacle  to  the  screen. 
Then  if  light  were  absolutely  rectilinear,  all  parts  of  the  screen 
above  Q  would  be  fully  illuminated,  all  points  below  in  com- 
plete shadow.  There  would  be  a  sharp  and  abrupt  division 
between  the  illuminated  part  and  the  shadow. 


BENDING  OF  LIGHT  INTO  SHADOW  35 

But,  as  a  matter  of  fact,  the  light  is  found  to  shade  off 
continuously,  though  rather  rapidly,  into  the  shadow;  and 
above,  the  point  Q  there  are  a  number  of  bright  and  dark 
bands,  parallel  to  the  slit  and  to  the  edge  of  the  obstacle.  The 
gradual  shading  off  of  the  light,  merging  into  the  shadow,  dis- 
proves the  rectilinear  propagation  of  light,  except  as  an  ap- 
proximation to  the  truth.  As  to  the  bands,  it  will  merely  be 
noted  here  that  they  are  readily  explainable  on  the  wave 
theory,  the  actual  explanation  being  deferred  to  later  pages. 
(See  section  70) 

Probably  the  crucial  reason  for  the  discarding  of  the 
corpuscular  theory  and  the  definite  adoption  of  the  wave 
theory  is  the  'following.  When  a  beam  of  light  strikes  the 
surface  of  glass  or  water  at  an  angle,  it  is  bent  toward  the 
normal,  that  is,  it  makes  a  more  acute  angle  with  the  perpen- 
dicular within  the  glass  than  in  the  air.  In  order  to  explain 
this,  the  advocates  of  the  corpuscular  theory  were  obliged  to 
assume*  that  the  corpuscles  are  attracted  by  the  glass  when 
they  get  very  close  to  it,  leading  to  the  conclusion  that  they 
travel  faster  in  the  glass  than  in  the  air.  On  the  other  hand, 
the  wave  theory  explains  this  bending  of  the  rays  at  the  sur- 
face very  easily  with  the  assumption  that  the  waves  travel 
slower  in  glass  or  water  than  in  air.  Thus  the  phenomenon  of 
refraction  furnishes  the  occasion  for  a  definite  clash  between 
the  two  theories,  one  demanding  that  light  travels  faster  in 
the  refracting  medium  than  in  air,  the  other  that  it  travels 
slower.  During  Newton's  life,  no  means  of  actually  measur- 
ing the  velocity  of  light  in  such  a  medium  as  glass  or  wa/ter5 
was(  known:  but  after  Foucault  had  devised  the  rotating 
mirror  method,  the  velocity  in  water  was  measured  by  filling 
a  long  tube,  fitted  with  glass  ends,  with  water,  and  inserting 
this  in  the  path  of  the  light.  The  experiment  showed  without 
any  possibility  of  doubt  that  light  travels  slower  in  water 
than  in  air. 

Since  this  experiment  definitely  discredits  the  corpuscular 
theory,  and  since  the  only  outstanding  objection  to  the  wave 
theory  is  the  hypothesis  of  the  ether,  for  whose  existence  we 
have  additional  evidence  from  electric  and  magnetic  phenom- 
ena, it  is  now  accepted  by  physicists  that  light  consists  of 


36  LIGHT 

very  short  waves  in  a  hypothetical  medium  extending  through 
all  space.  Even  when  light  passes  through  a  material  like 
glass  or  water,  we  regard  the  ether  as  the  carrier  of  the  waves. 
"We  may  think  of  the  molecules  of  the  glass  as  existing  in  the 
ether.  Their  presence  modifies  the  transmission  of  the  waves, 
partly  by  changing  the  wave-velocity,  partly  by  absorbing 
part  of  the  wave-energy,  which  they  change  into  heat-energy, 
for  heat  is  always  produced  when  light  is  absorbed.  A  rough 
image  of  the  state  of  affairs  may  be  gotten  by  considering 
waves  on  the  surface  of  water  on  which  are  floating  many 
pieces  of  wood.  The  water  represents  the  ether,  the  pieces  of 
wood  the  atoms  or  molecules  of  the  material  substance. ' 

19.  Nature  of  the  ether. — It  is  difficult  to  speculate  about 
the  nature  of  the  ether.  There  aret  no  reasons  for  believing 
that  it  is  atomic  or  molecular  in  structure,  and  it  seems  to 
offer  absolutely  no  resistance  to  the  passage  of  bodies  through 
it.  A  philosophically  minded  person  might  ask  the  question: 
Just  what  do  we  mean  when  we  say  there  is  an  ether!  Such 
a  question  is  worth  while  because  it  leads  us  to  take  stock  of 
our  knowledge;  but  probably  the  only  answer  that  can  be 
given  i is  this;  if  the  wave  theory  of  light  is  true,  (and  it  is 
supported  by  too  many  facts  now  for  us  to  doubt  it)  then  the 
statement  that  there  is  an  ether  means  only  that  empty  space 
has  properties  other  than  mere  extension,  properties  that  enable 
disturbances  carrying  energy  to  pass  through  it,  the  passage 
requiring  finite/  time.  Whether  we  say  there  is  an  ether,  or 
that  empty  space  has  properties  other  than  those  of  pure 
geometry,  matters  little,  but  the  name  "ether"  is  a  convenient 
one  to  symbolize  these  properties,  and  we  shall  hereafter  use 
it  in  this  sense. 

We  shall  see  that  it  is  comparatively  easy  to  devise  ex- 
periments for  measuring  the  wavelength  of  light,  and  that  the 
measurements  can  be  carried  out  to  a  very  high  degree  of 
precision.  (Chapter  Yir  and  IX)  It  will  be  easy  to  explain 
the  laws  of  refraction  from  the  fact  mentioned  above,  that  the 
velocity  of  light  is  less  through  such  a  substance  as  glass  than 
through  air.  somewhat  less  through  air  than  in  the  free  ether. 
(Chapter  IV)  "We  can  explain  the  formation  of  a  spectrum  by 
a  prism,  by  showing  that  through  glass  the  velocity  of  shorter- 
waves  is  less  than  that  of  longer  ones. 


WAVES  IN  GENERAL  37 

But  when  all  this  done,  it  must  not  be  forgotten  that  we 
shall  still  be  much  in  the  dark  as  to  many  important,  questions. 
In  the  first  place,  are  these  waves  longitudinal,  like  sound 
waves,  or  transverse,  like  those  of  a  plucked  string?  Are  they 
purely  mechanical  waves,  or  do  they  consist  of  rapidly  alter- 
nating electrical  disturbances,  or  are  they  disturbances  of  a 
kind  unrelated  to  any  other  physical  phenomena?  What  is 
the  nature  of  the  phenomena  going  on  within  the  atoms  of  a 
substance  which  is  emitting  light,  or  in  those  of  a  substance 
which  is  absorbing  it?  How  and  why  does  the  presence  of 
material  molecules,  in  such  a  substance  as  glass  or  water,  change 
the  velocity  of  the  waves  in  the  ether  surrounding  and  per- 
meating them? 

Some  of  these  questions  have  been  solved  satisfactorily, 
some  have  been  only  partially  solved,  others  are  still  open.  It 
is  clear  that  if  we  are  ever  able  to  answer  all  of  them  we  shall 
know  a  great  deal  about  the  inner  structure  of  tHe  atom  and 
the  molecule;  and  since  the  acquisition  of  such  information  is 
one  of  the  principal  aims  of  physical  research,  the  study  of 
light  becomes  one  of  the  most  important  branches  of  science. 

20.  Waves  in  general.  Plane  waves. — Before  going  on  with 
the  study  of  light  as  a  wave-motion,  we  shall  devote  some  space 
to  the  consideration  of  waves  in  general.  We  find  cases  of 
waves  which  advance  (1)  in  only  a  single  dimension,  like 
those  that  travel  along  a  stretched  rope  if  it  is  struck  or  moved 
in  any  way.  (2)  in  two  dimensions  (i.  e.,  spreading  over  a  sur- 
face) like  water  waves,  and  (3)  in  three  dimensions,  like  sound 
waves,  light  waves,  or  the  waves  that  would  spread  through  a 
block  of  jelly  if  some  point  in  the  interior  were  set  vibrating. 
In  the  last  case,  it  is  clear  that  the  waves  would  spread  out  in 
spheres  with  the  point  of  origin  as1  center,  the  direction  of 
advance  being  along  the  radii.  Under  certain  circumstances, 
however,  we  could  have  cases  where  they  advance  in  planes, 
the  direction  of  advance  being  perpendicular  to  these  planes. 
There  would  be  an  approximation  to  this  condition,  for  exam- 
ple, at  a  very  great  distance  from  the  point  of  origin,  for  a 
small  section  of  a  sphere  of  very  great  radius  is  nearly  plane; 
but  such  waves  would  be  feeble  because  the  energy  initially 
given  to  them  would  be  spread  over  a  great  surface.  In  the 
case  of  light,  as  we  shall  see  in  Chapter  V,  such  plane  waves 


38  LIGHT 

can  be  produced  by  the  use  of  mirrors  or  lenses,  without  such 
a  weakening  of  intensity. 

The  nature  of  plane  waves 
may  be  understood  by  imagining 
a  block  of  jelly  as  in  figure  17,  to 
one  face  of  which  is  attached  a 
rigid  board  that  is  moved  har- 
monically back  and  forth  in  its 
own  plane,  in  the  direction  of  the 
Figure  17  line  AB.  At  any  instant,  each 

point  in  a  plane  parallel  to  the  board  will  be  in  the 
same  condition  of  motion,  or,  as  we  say,  in  the  same  phase. 
lln  general,  any  such  surface,  every  point  of  which  is  in  the 
same  phase,  is  called  a  wave-front,  whether  it  is  at  the  begin- 
ning of  a  train  of  waves  or  not.  In  figure  17,  every  plane  in 
the  jelly  parallel  to  the  board  is  a  wave-front,-  and  in  the  case 
of  waves  coming  from  a  point-source,  every  sphere  with  its 
center  at  the  source  is  a  wave-front. 

We  may  have  cases  of  a  single  wave,  like  the  sound-wave 
sent  out  from  an  explosion,  and  also  cases  where  there  is  a 
train  of  waves,  such  as  those  sent  out  by  a  tuning-fork.  The 
shape  of  the  waves  may  be  simple  or  complicated.  For  any 
medium  through  which  waves  of  all  lengths  travel  with  the 
same  velocity  (as  is  true  for  the  free  ether  with  light-waves) 
a  disturbance  of  any  kind  will  travel  onward  without  changing 
its  shape,  whatever  that  shape  may  be.  For  example,  by  giv- 
ing the  board  of  figure  17  a  suitable  motion,  waves  of  the  form 
shown  in  figure  18  could  be  made  to  travel  through  the  jelly. 
A  mathematical  theorem  due  to  Fourier  proves  that  any 
such  periodic  form  can  be  made  up  of  a  series  of  simple  sine 


Figure    18 

and  cosine  forms,  of  different  wavelengths.  For  this  reason, 
we  are  compelled  to  make  a  special  study  of  waves  of  these 
simple  forms. 


FORMULA  FOR  A  WAVE 


39 


Figure    19 

21.   Mathematical  formula  for  a  wave. — In  figure  19,  A 
represents  a  sine-curve,  whose  equation  is 

ir   -    2?rX  /i\ 

y  =  K.sm —  (1) 


B  a  cosine-curve,  whose  equation  is 


TT 

y  =  K.cos  — 
a 

and  C  a  third  curve  whose  equation  is 

y  =  K.COK  —  (x  -.  —  a) 
a 


(2) 


(3)    / 


Evidently,  all  three  are  exactly  the  same  in  form,  and  either 
can  be  transformed  into  one  of  the  others  by  shifting  it  along 
the  axis  of  x.  In  fact,  we-  can  represent  either  of  the  three 
curves  by  the  formula  (3)  provided  a  is  given  a  suitable  value, 
different  in  each  case.  If  a  =  0,  we  have  the  simple  cosine 
curve.  If  a  —  a/4  we  have 


T2,  27T    f  a\  /27TX  7T\  2?rX 

y  =  K.cos  —  (x  —  -    =  K.COS     ---  -  )  —  K.sm  — 
a  4  /  \  a        2  /  a 

the  equation  of  the  sine  curve.  Whatever  value  a  may  have, 
its  presence  indicates  that  equation  (3)  represents  a  simple 
cosine  curve  shifted  in  the  positive  direction  of  x  by  the  dis- 
tance a.  For  y  has  the  same  value  for  x  in  (3)  that  it  has 
for  x  —  a  in  (2). 

Equations  (1),  (2),  and  (3)  do  not  represent  waves,  but 
only  stationary  curves.  For  such  a  curve  to  become  a  wave, 
it  must  progress  steadily  to  the  right  (or  left).  Since  the 
symbol  a  indicates  a  shift  to  the  right,  (3)  can  be  changed 
to  represent  a  wave  instead  of  a  stationary  curve  if  a  is  re- 


40  LIGHT 

placed  by  a  term  containing  the  time,  which  term  indicates  the 
movement.  If  t  represents  the  time  and  V  the  velocity  of  the 
waves,  then  in  t  seconds  the  curve  must  be  shifted  along  the 
x  axis  a  distance  Vt.  Therefore,  putting  Vt  for  a,  the  wave 
equation  is 

y  —  K.cos-  (x  —  Vt)  (4) 

a 

a  is  called  the  wavelength,  for  it  is  evident  from  figure  19  that 
it  is  the  distance  from  crest  to  crest,  or  from  trough  to  trough. 
The  quantity  K,  called  the  amplitude,  gives  the  maximum 
value  y  can  have.  Equation  (4)  is  an  equation  of  three  varia- 
bles, y,  x,  t.  x  and  t  are  called  independent  variables,  y  the 
dependent  variable.  The  equation  gives  the  value  of  y  for  any 
stated  distance  x  from  the  origin  and  for'  any  stated  time  t. 

It  is  often  desirable  to  introduce  another  constant  term, 
c,  within  the  parenthesis,  making  the  equation  read 

y  =  K.cos—  (x  —  Vt  —  e)       (5) 
a 

The  only  difference  between  (4)  and  (5)  is  that  the  former 
represents  a  wave  which,  at  time  zero,  has  the  position  B  of 
figure  19,  while  the  latter  is  one  which  at  that  instant  has  the 
position  A.  B,  C.  or  any  other,  depending  upon  the  value  of  e. 
This  quantity  e  is  called  the  pilose-constant  ,  the  whole  quan- 
tity whose  cosine  is  to  be  taken,  viz., 


a 
being  known  as  the  phase. 

A  wave  advancing  to  the  left  would    be    represented    by 
.equation  (4)  or  (5)  with  the  sign  of  V  changed;  thus, 

y  =  K.cos  2-(x  +  Vt  —  c) 
a 

So  far  we  have  regarded  y  as  a  real  mechanical  displace- 
ment, at  right  angles  to  the  direction  in  which  the  waves  are 
propagated.  In  the  case  of  transverse  waves  in  a  string,  it 
is  indeed  just  that.  The  same  thing  is  true  of  the  mechanical 
waves  in  the  block  of  jelly,  illustrated  in  figure  17,  and  in  all 
such  eases  the  curves  of  figure  19  give  a  true  picture  of  the 
contour  of  the  waves  at  different  times. 


INTERFERENCE  41 

But  there  are  cases  of  wave-motion  ins  which  this  is  not 
the  case.  „  For  example,  if  a  stiff  rod  or  a  stretched  metal  wire 
be  stroked  longitudinally  with  a  rosined  piece  of  leather, 
longitudinal  waves  are  set  up  in  which  the  displacement  is 
parallel  to  the  direction  of  propagation  of  the  waves.  In  such 
a  case  we  may  still,  for  convenience,  plot  y  at  right  angles  to 
x,  but  the  graph  so  obtained  is  only  a  graph,  and  not  a  true 
picture  of  the  wave  contour. 

It  may  be  that  a  wave  does  not  even  consist  of  mechanical 
vibrations  at  all.  There  are  cases,  for  instance,  of  temperature 
waves,  in  which  alternations  of  temperature,  above  and  below 
a  mean  value,  are  sent  through  a  material.  It  is  quite  con- 
ceivable also  that  waves  should  exist  which  consist  of  electric 
disturbances,  for  instance  regions  in  which  there  is  an  electric 
intensity  directed  upward,  separated  by  those  in  which  it  is 
directed  downward,  these  alternating  regions  following  one 
another  through  space  with  great  rapidity.  Since  we  have  no 
certain  knowledge  that  such  electric  states  in  the  ether  are 
necessarily  accompanied  by  any  real  motion,  of  the  ether  or  of 
anything  else,  the  quantity  y  in,  such  a  case  would  have  to 
represent  the  intensity  of  the  electric  field  at  the  distance 
represented  by  x  and  the  time*  represented  by  t.  It  will  be 
shown  later  later  (Chapter  XIV)  that  such  electric  waves 
actually  exist,  that  the  waves  of  wireless  telegraphy  are  un- 
doubtedly such,  and  that  we  have  convincing  evidence  that 
light  waves  are  also  of  the  same  nature,  differing  from  the 
waves  of  wireless  only  in  length. 

But  many  of  the  phenomena  of  waves,  such  as  interfer- 
ence, diffraction,  and  some  of  the  phenomena  of  reflection  and 
refraction,  would  be  the  same  no  matter  what  the  nature  of 
the  disturbance  might  be;  and  therefore  it  will  be  convenient, 
for  the  time  being,  to  think  of  light  waves  as  if  they  were 
really  waves  of  mechanical  displacement.  Whether  they  are  to 
be  thought  of  as  transverse,  like  those  in  a  plucked  string,  or 
longitudinal,  like  those  of  sound,  need  not  be  considered  yet, 
but  evidence  will  be  produced  in  Chapter  XII  to  help  us  de- 
cide between  these  alternatives. 

22.  Interference.  Fresnel's  mirrors. — One  of  the  most 
convincing  proofs  of  the  wave  theory  of  light  is  the  phenom- 


42  LIGHT 

enon  of  interference,  in  which  two  separate  beams  of  light 
annul  one  another  at  certain  places,  producing  darkness,  and 

at  other  places  produce  a 
brightness  much  greater 
than  either  alone  could 
cause.  The  theory  of  this 
s,  m  -  phenomenon  is  as  follows : 

sz  m*    ,  Suppose  we  have  two 

sources  of  light,  Si  and  S2, 
figure  20,  in  the  form  of 
narrow    slits    or  round 
holes,    through    each    of 
which   comes   light   of   ex- 
actly the  same  wavelength.*     At  first,  we   shall  also  suppose 
that  the  two  pencils  of  light  are  in  phase,  that  is,  that  when- 
ever a  crest  starts  from  one  a  crest  will  also  be  starting  from 
the  other.     This  means  that,  if  we  should  write  a  formula  of 
the  type 

yrzrK.COS— '     (X  —  Vt  —  c) 

a 

for  each  pencil,  €  would  have  the  same  value  in  both.  It  will 
be  easier  however  to  discuss  this  case  without  the  use  of  the 
formulae. 

The  light  from  each  slit  falls  on  the  white  screen  AB,  and 
we  shall  first  investigate  what  happens  at  that  point,  C,  which 
is  equally  distant  from  the  two  slits.  Whenever  a  crest  reaches 
C  from  St  a  crest  will  also  reach  it  from  S2,  and  similarly  a 
trough  from  St  and  a  trough  from  S2  will  reach  C  at  the  same 
instant.  Consequently,  the  amplitude  of  the  vibrations  at  C 
will  be  double  that  which  would  exist  if  light  from  only  one 
slit  reached  it,  and  the  screen  will  be  very  bright  there.  There 
will  be  other  points  on  the  screen  for  which  the  same  statement 

*It  is  customary  among  physicists  to  speak  sometimes  of  a  small 
hole  or  slit  through  which  light  is  passed  as  the  source  of  the  light, 
although  in  fact  the  real  source  is  a  flame,  a  spark,  an  arc-lamp,  or 
perhaps  the  sun.  This  real  source  is  placed  close  to  the  slit  or  hole, 
or  else  an  image  of  it  is  thrown  upon  the  latter  by  the  use  of  a  lens 
or  mirror.  The  object  of  the  slit  or  hole  is  simply  to  provide  a  very 
narrow  opening  for  the  light  to  come  through.  If  the  source  proper  is 
itself  small  enough,  the  slit  may  be  dispensed  with. 


INTERFERENCE  43 

holds  true.  For  instance,  if  M!  is  the  point  just  one  wave- 
length nearer  to  St  than  to  S2,  and  M\  the  point  one  wave- 
length nearer  to  S2  than  to  Sx,  at  each  of  these  points  .crests 
will  arrive  together  from  the  two  slits,  and  also  troughs  will 
arrive  together,  and  therefore  these  too  will  be  points  of  bright- 
ness. The  same  may  be  said  of  M2,  which  is  two  wavelengths 
nearer  to  S1  than  to  S2,  and  of  any  point  on  the  screen  which 
is  an  exact  whole  number  of  wavelengths  nearer  to  one  of  the 
slits  than  to  the  other. 

On  the  other  hand,  consider  the  points  mlt  m'lt  m,.  m'2, 
etc.,  each  of  which  is  so  situated  that  it  is  either  %  wavelength, 
•%  wavelength,  or  in  general  any  odd  number  of  half -wave- 
lengths, nearer  to  one  slit  than  to  the  other.  Each  of  these 
points  will  receive  a  crest  from  one  slit  at  the  same  time  as  it 
receives  a  trough  from  the  other.  In  other  words,  the  pencils 
of  light  coming  from  the  two  sources  will  at  these  points  be 
opposite  in  phase,  so  that  they  will  annul  one  another,  and  the 
points  will  be  dark.  It  need  hardly  be  pointed  out  that  there 
will  be  points,  such  as  one  between  C  and  m^  which  will 
neither  be  as  bright  as  C  nor  completely  dark  as  at  m1?  since 
here  the  two  pencils  of  light  meet  neither  exactly  in  phase  nor 
exactly  opposite  in  phase.  In  fact,  a  moving  point,  going  from 
C  up  toward  A,  would  first  be  in  intense  illumination,  which 
would  fade  out  to  darkness  at  m1?  then  brighten  again  to  a 
maximum  at  M,,  fade  to  darkness  again  at  m2,  etc. 

We  should  expect  then,  according  to  the  wave  theory,  to 
find  a  number  of  bright  regions,  separated  by  dark  regions. 
Tn  fact,  these  should  be  drawn  out  into  bright  and  dark 
streaks,  or  fringes,  as  they  are  called,  perpendicular  to  the 
pl'ane  for  which  the  figure  is  drawn.  For,  even  if  the  slits, 
had  no  appreciable  length  in  this  direction,  the  loci  of  bright 
or  dark  regions  would  still  be  drawn  out  into  lines. 

Now  let  us  see  how  all  this  would  be  altered  if  the  two, 
pencils  of  light  were  not  exactly  in  phase  as  they  came  through 
the  slits.  We  should  still  expect  to  have  fringes  but  they 
would  not  occur  at  quite  the  same  place  on  the  screen.  For1 
instance,  if  the  difference  in  phase  were  such  that  a  crest  would 
start  from  Si  and  a  trough  from  S2  at  the  same  instant,  (and 
vice  versa  of  course)  then  the  places  we  have  marked  as  bright 


44  LIGHT 

would  be  dark,  and  those  we  have  marked  dark  would  be 
bright.  Herein  lies  a  certain  difficulty  in  subjecting  these 
predictions  to  experimental  proof.  The  trouble  is  not  so  much 
to  keep  the  two  pencils  in  the  same  phase,  since  we  don't  much 
care  which  points  are  dark  and  which  bright,  so  long  as  tihe 
fringes  stay  steady  long  enough  for  us  to  see  them.  But  they 
will  not  keep  steacty  unless  the  two  pencils  at  least  keep  the 
same  relation  to  one  another  in  phase,  and  this  they  will  not 
do  unless  they  were  originally  part  of  the  same  pencil  or  beam, 
that  is,  unless  they  originated  in  the  same  ultimate  source.  It 
must  not  be  supposed  that  any  source  of  light  is  perfectly 
steady.  "We  might  think  of  it  as  sending  out  a  regular  train 
of  waves  perhaps  a  meter  in  length,  followed  by  a  break  and 
another  train,  perhaps  longer,  perhaps  shorter,  there  being  no 
fixed  relation  between  the  phase-constant  in  the  first  train  and 
that  in  the  second.  In  other  words,  the  light  comes  in  bunches 
of  waves,  rather  than  in  a  long  uninterrupted  series.  Now,  if 
changes  of  this  sort  are  going  on  in  the  light  coming  from  each 
slit,  and  the  breaks  are  occurring  quite  independently  in  the  two 
pencils,  it  is  evident  that,  although  fringes  would  be  present 
on  the  screen  at  any  instant,  the}^  woiild  shift  their  position 
with  every  break  in  either  pencil.  If  we  assume  that  the  aver- 
age length  of  an  uninterrupted  train  of  waves  is  one  meter, 
then  since  light  travels  300,000,000  meters  per  second,  there 
would  be  at  least  300,000,000  shifts  per  second  in  the  positions 
of  the  fringes.  Consequently,  no  fringes  could  be  seen,  and 
the  screen  would  appear  uniformly  illuminated. 

Therefore,  in  order  to  see  such  fringes,  it  is  absolutely 
necessary  to  get  two  pencils  that  have  the  same  origin,  so  that 
whenever  the  phase  of  one  pencil  suddenly  changes,  that  of 
the  other  will  undergo  the  same  change.  There  are  several 
ways  of  accomplishing  this  end,  the  most  satisfactory  being 
one  due  to  the  French  physicist  Fresnel,  a  diagram  of  which 
is  shown  in  figure  21.  He  used  only  one  slit,  S,  illuminated  by 
any  source  of  light,  but  he  allowed  the  beam  from  S  to  fall  on 
two  mirrors  Mt  and  M2,  very  slightly  inclined  to  one  another, 
each  of  which  reflected  light  to  the  screen.  The  light  strikes 
the  screen  exactly  as  if  it  came  from  the  two  images  SL  and  S2, 
which  may  be  thought  of  as  replacing  the  two  independent 


THE  FRESNEL  MIRRORS 


45 


slits  St  and  S2  of  figure  20,  with  the  one  important  differ- 
ence that  now  whatever  change  occurs  in  one  pencil  will  also 
occur  at  the  same  instant  in  the  other.  Thus  they  will  always 
have  the  same  relation  to  one  another  in  phase,  and  the  fringes 
will  be  steady  and  therefore  visible. 


Figure    21 


Figure   22 

This  experiment  works  very  satisfactorily,  though  the  ad- 
justment is  somewhat  difficult,  since  the  slit  must  be  very 
accurately  parallel  to  the  line  in  which  the  planes  of  the  two 
mirrors  intersect.  Figure  22  is  a  photograph  of  fringes  taken 
by  this  method.  The  light  used  was  the  violet  of  a  definite 
wavelength  coming  from  the  "mercury-arc." 

Referring  back  to  figure  21,  let  d  represent  the  distance 
between  the  two  apparent  sources  St  and  S2,  D  the  distance 
from  their  plane  to  the  plane  of  the  white  screen,  C  the  point 
equally  distant  from  Si  and  S2.  Let  P  be  any  point  on  the 
screen,  within  the  plane  of  the  diagram,  and  x  its  distance 
from  C.  We  shall  first  calculate  what  values  x  may  have  in 
order  for  P  to  be  one  of  the  points  of  maximum  brightness, 
and  from  this  result  find  the  distance  between  centers  of  the 
bright  fringes. 

If  L  represents  the  difference  between  the  distances  SXP 
and  S2P,  P  will  be  a  point  of  maximum  or  minimum  illumina- 
tion according  as  L  is  equal  to  an  even  or  an  odd  number  of 


46  LIGHT 


half-wavelengths.     Therefore  our  first  step  will  be  to  express 
L  in  terms  of  measurable  quantities  d,  D,  and  x.     By  simple 


geometry, 


Therefore, 


L  =  S2P  —  S,P  =  \/D2  +  x2  +^  -f  xd  —  Vo2  +  x2  +  I2._xd 

4  4 

For  convenience,  we  shall  represent  the  quantity  D2  +  x2  +— 

4  9 

which  appears  in  both  radicals,  by  a  single  term  D02.     Then 


L  =  Vl>o2  +  xd  —  VD02  —  xd 

These  two  simple  radicals  can  be  expanded  in  series  form,  by 
using  the  binomial  theorem,  or  applying  the  ordinary  rules 
for  the  extraction  of  the  square-root.  The  results  are 


n  —  T  xd         x2d2    ,   x3ds 

+  xd  =  D.,  +  —  -  _  +*—-  etc. 

S  —  n         —  -     x2(P       x3d3 
-  Pt--  2D0       8D03     16D05  " 

Subtracting  the  lower  from  the  upper,  we  get 

xd        x3d3 

=  i£+W4 

In  practise,  D  is  about  the  order  of  100  cm.,  d  about  half  a 
millimeter,  or  .05  cm.,  and  x  of  course  has  various  values,  of 
which  the  greatest  will  perhaps  be  1  cm.  If  we  substitute 
these  values,  we  see  that  in  the  first  place  Dy  will  not  differ 
from  D  itself  by  more  than  about  1/20000  cm.,  so  that  D  may 
be  substituted  for  D0.  Furthermore,  the  value  of  the  first 
term  in  the  last  equation  comes  out  to  be  about  .0005,  the  next 
term  .0000000000015,  and  the  succeeding  terms  still  more  minute. 
Of  course,  D,  d,  and  x  need  not  have  exactly  the  values  here 
assumed,  but  the  illustration  suffices  to  show  that  in  any  case 


THE  FRESNEL  MIRRORS  47 

D  is  so  nearly  equal  to  D0  that  the  difference  is  negligible,  and 
that  in  the  final  expression  for  L  only  the  first  term  is  large 
enough  to  measure.  Consequently,  we  shall  make  no  error 
greater  than  the  unavoidable  errors  of  measurement  if  we 
adopt  as  the  correct  vaiue  for  L 

L  =  xd/D 

Using  the  letter  \  to  represent  the  wavelength  of  the  light, 
we  must  have,  in  order  that  P  may  be  a  bright  point,  that  L 
takes  one  or  other  of  the  values  0,  ±\,  ±2\,  —  3A,  etc.  Or, 
since  x  —  DL/d,  P  is  a  point  of  maximum  brightness  when  x 
has  the  value  0,  ±D,\/d,  ±2Dx/d,  etc. 

This  shows  us  that  the  fringes  are,  at  least  approximately, 
equally  spaced,  the  distance  from  center  to  center  being  Dx/d. 
This  distance  can  be  measured  with  some  degree  of  accuracy, 
and  also  the  distance  D.  The  remaining!  distance  d  is  more 
difficult  to  measure,  partly  because  it  is  smaller,  and  partly 
because  it  is  not  the  distance  between  real  slits,  but  between 
two  images.  However,  it  can  be  measured  by  indirect  means, 
and  then  everything  necessary  is  known  in  order  to  calculate 
the  wavelength.  It  is  found,  as  we  should  expect,  that  the 
wavelength  as  so  determined  depends  upon  the  kind  of  light 
used.  If  only  deep  red  light  enters  the  slit,  the  width  of  the  re- 
sulting red  fringes  indicates  the  wavelength  to  be  about  .00007 
cm. ;  while  if  deep  violet  is  used  the  fringes  are  only  a  little 
miore  than  half  as  far  apart  as  the  red,  indicating  the  wave- 
length of  this  color  to  be  about  .000038  cm.  The  other  colors 
have  wavelengths  between  these  two  extremes.  But  the  ex- 
tremes themselves  are  not  very  definite,  since  it  is  found  that 
some  people  can  see  deeper  red  or  deeper  violet  than  others. 
This  fact  makes  us  suspect  the  existence  of  wavelengths  longer 
than  the  red  or  shorter  than  the  violet,  to  which  nobody's 
eyes  are  sensitive.  We  shall  find  later  that  there  are  such 
waves.  (Sections  64  to  67) 

Although  this  interference  experiment  gives  us  a  means  of 
measuring  the  wavelength  of  light,  it  is  not  an  accurate  method. 
More  complicated  interference  experiments,  to  be  described 
later,  allow  us  to  measure  wavelengths  with  an  accuracy  of 
1/1000  of  1%.  and  in  a  few  cases  the  precision  has  been  car- 
ried even  farther. 


48  LIGHT 

23.  Interference  in  white  light. — Let  us  consider  what 
would  happen  if  white  light,  instead  of  light  of  only  one  color, 
were  admitted  through  the  slit  in  figure  21.  Evidently  every 
wavelength  would  produce  its  own  set  of  fringes,  the  spacing 
being  different  for  each  wavelength,  and  there  would  be  much 
overlapping  of  fringes  of  different  color.  Only  one  point,  the 
point  C.  would  be  bright  for  all  colors,  since  it  is  equally  dis- 
tant from  the  two  slit-images.  The  central  fringe  would  there- 
fore be  white.  But  the  first  red  fringe  on  either  side  of  the 
center  would  be  slightly  farther  away  than  the  first  violet 
fringe.  Consequently,  the  totality  of  each  of  these  two  fringes 
would  be  composed  of  fringes  due  to  all  the  different  wave- 
lengths, for  no  two  of  which  would  the  maxima  come  in  ex- 
actly the  same  place,  violet  being  on  the  side  nearest  to  C, 
red  on  the  other  side.  One  might  regard  each  of  these  fringes 
as  a  very  short  and  impure  spectrum.  It  would  be  white  near 
its  middle,  with  a  violet  inner  edge  and  a- red  outer.  This 
effect  would  be  more  pronounced  for  the  second  fringe  on  each 
side,  still  more  for  the  third,  etc.,  as  if  each  succeeding  fringe 
were  a  longer  and  longer  spectrum.  At  the  distance  of  two 
or  three  fringes  away  from  0,  they  begin  to  seriously  overlap ; 
and  at  the  distance  of  six  or  eight,  the  overlapping  becomes 
so  complex  that  all  color-effect  is  lost,  the  fringes  are  no  longer 
visible,  and  the  screen  becomes  uniformly  white.  When  white 
light  is  used,  therefore,  only  a  small  number  of  fringes  are 
ever  visible,  whereas  with  light  of  a  single  wavelength,  spoken 
of  as  monochromatic  light,  a  great  number  may  be  seen,  pro- 
vided the  two  beams  overlap  over  a  sufficiently  extended 
region.  A  further  discussion  of  interference  in  white  light 
will  be  found  in  section  82. 

Problems. 

1.  If,  in  Foucault's  rotating  mirror  experiment,  figure  7, 
a  cylinder  of  some  material,  in  which  the  velocity  of  light  is 
less  than  in  air,  is  inserted    between  the  mirrors  Mt  and  M2, 
what  would  be  the  effect  upon  the  distance  f^?    What  would 
be  the  effect  if  the  material  were  inserted  between  Mx  and  L? 

2.  Plot  to  scale,  on  the  same  diagram,  the  curves  repre- 
sented   by    formulae    (2)    and    (3),    letting    K  =  1,  a  =  4, 


THE  FRESNEL  MIRRORS  49 

a  =  1.5.  (It  will  suffice  to  plot  only  the  peaks  and  troughs, 
and  the  points  where  the  curves  cross  the  x-axis)  Show  by 
scaling  that  the  curve  (3)  lies  to  the  right  of  (2)  by  the  amount 
predicted  in  the  text. 

3.  Plot  the  curve  for  equation    (4),    giving    any    desired 
value  to  the  time. 

4.  If  the  two  slit-images  of  the  Fresnel  Mirror  experiment 
are  l/10mm.  apart,  how  far  away  must  the  screen  be  to  have 
fringes  3mm.  apart,  if  the  wavelength  is  .00005cm.? 

5.  Suppose  the  light  coming  from  the  two  sources  of  figure 
20  had  not  the  same  wavelength.    What  would  be  the  result? 


CHAPTER  IV. 

24.  Reflection  and  refraction.  Huyghens'  principle.  Index  of  re- 
fraction.— 25.  Total  reflection.  Critical  angle. — 26.  Deviation  through 
a  prism. 

24.  Reflection  and  refraction. — Huyghens'  principle.  In- 
dex of  refraction. — The  laws  of  reflection  and  refraction,  so 
far  as  concerns  only  the  direction  of  the  reflected  and  re- 
fracted waves  and  not  their  intensity,  are  easy  to  derive  by  an 
application  of  geometry  to  the  wave  theory. 


Figure i 23 

Let  MN,  in  figure  23,  represent  the  surface  of  a  sheet  of 
water,  or  the  plane  and  polished  surface  of  glass  or  some  other 
reflecting  and  refracting  medium.  We  shall  suppose  the  water, 
glass,  or  other  material  to  fill  the  space  below  MN,  the  medium 
above  being  the  free  ether.  Plane  waves  are  advancing  through 
the  ether,  in  the  direction  indicated  by  the  arrow  P,  toward 
the  surface.  The  lines  ab,  ajbj,  etc.,  represent  successive  posi- 
tions of  an  advancing  wavefront,  as  it  approaches  MN.  Our 
problem  is  to  determine  the  position  of  the  reflected  wave  and 
the  refracted  wave  to  which  this  incident  wave  gives  rise. 

(50) 


HUYGHENS'  PRINCIPLE  51 

In  order  to  do  this  we  shall  make  use  of  a  principle 
enunciated  by  the  Dutch  physicist  Huyghens,  which  applies  to 
all  types  of  waves.  It  may  be  stated  as  follows:  A  wave-front 
propagates  itself  by  virtue  of  the  fact  that  each  point  in  the 
medium,  as  the  wave-front  reaches  it,  becomes  itself  a  center 
of  disturbance  from  which  a  spherical  wave  is  sent  out;  and 
the  further-advanced  position  of  the  original  wavefront  is 
nothing  more  nor  less  than  the  envelope  of  all  the  secondary 
wavelets  sent  out  from  the  totality  of  points  taken  as  centers. 

"When  the  wavefront  ab  reaches  the  position  a^b.,,  the  point 
a2  therefore  becomes  the  center  of  such  a  spherical  wavelet, 
not  only  in  the  medium  above  MN,  but  also  in  that  below. 
But  the  wavelets  in  the  two  media  will  not  advance  equally 
fast,  because  the  velocity  of  light  is  less  in  the  lower  medium 
than  in  the  free  ether.  Let  us  suppose  the  velocity  in  the 
lower  medium  to  be  1/n  of  that  in  the  upper.  Then,  while 
the  incident  wavefront  is  travelling  onward  from  the  position 
a2b.,  till  it  reaches'  the  reflecting  surface  at  b3x  the  secondary 
wavelet  from  a2  will  have  acquired  a  radius  equal  to  b2b3  in 
the  upper  medium,  but!  a  radius  of  only  1/ri  of  b2b3  in  the 
lower.  Therefore  an  arc  is  drawn  in  the  upper  medium  with 
radius  b2b3,  and  one  in  the  lower  with  radius  b2b3/n,  both  hav- 
ing a-,  as  center.  What  has  occurred  at  a2  will  occur  at  every 
point  on  MN  as  the  advancing  incident  wavefront  reaches  it, 
except  that,  if  we  want  to  construct  the  reflected  and  refracted 
wavefronts  for  the  time  when  the  incident  wave  is  at  b3,  we 
must  take  the  radii  of  the  secondary  wavelets  shorter  and 
shorter  for  centers  nearer  and  nearer  to  b3.  Thus,  for  the 
point  c  as  center,  we  take  the  radius  equal  to  mb,,  in  the  upper 
medium,  mb,/n  in  the  lower.  For,  when  the  incident  wave- 
front  has  reached  c  it  .has  also  reached  m,  and  still  has  tine 
distance  mb:,  to  travel.  For  d  as  center,  the  proper  radii  are 
obc  and  ob3/n,  and  so  on.  There  should  be  an  infinite  number 
of  such  secondary  wavelets,  of  which  only  a  few  are  drawn  in 
the  figure.  A  plane  passing  through  b3  and  tangent  to  all  the 
secondary  wavelets  in  the  upper  medium  gives  the  wavefront 
of  the  reflected  light,  and  another  through  b3  tangent  to  all 
those  in  the  lower  medium  gives  that  of  the  refracted  light. 
Each  of  these  advances  perpendicular  to  its  own  plane,  as 


52  LIGHT 

shown  by  the  arrows  Q  and  R  respectively.  The  student  may 
ask  what  becomes  of  those  parts  of  the  secondary  wavelets 
which  do  not  lie  on  the  common  tangent  plane.  It  can  be 
shown  that  they  mutually  annul  one  another  by  interference, 
if  we  take  account  not  only  of  the  crests  of  waves  but  also  of 
the  troughs. 

The  angle  b2a2b3,  which  is  a  dihedral  angle  between  the 
reflecting  surface  and  the  plane  of  the  incident  wavefront,  is 
called  the  angle  of  incidence.  The  dihedral  angle  between  the 
the  reflecting  surface  and  the  reflected  wavefront,  a2b3a3,  is 
the  angle  of  reflection,  and  that  between  the  reflecting  surface 
and  the  refracted  wavefront,  a2b3a4,  is  the  angle  of  refraction. 
a.a,  is  drawn  perpendicular  to  the  reflected  wavefront,  a2a4 
perpendicular  to  the  refracted  wavefront.  Since  a2a3  is  equal 
to  b2bo,,  the  triangles  a2a3b3  and  b2b3a2  are  equal,  and  the  angles 
of  incidence  and  reflection  are  equal.  The  triangles  b2b3a2  and 
a^bg  are  not  equal,  but  are  both  right-angled  triangles,  the 
angles  at  b2  and  a4  being  each  equal  to  90°.  Therefore,  repre- 
senting1 the  angle  of  incidence  by  i  and  the  angle  of  refraction 
by  r,  we  have 

b2b- 

sin.  i  =  -v^ 


a, a, 

sin.  r  =  ^~ 
Sjb, 

Therefore, 

sin.  i  _    bab,;  _  ^ 
sin.  r    "   a2a4 

By  the  method  of  constructing  the  figure,  a2a4,  bears  the 
same  relation  to  b2b3  that  the  velocity  of  light  in  the  lower 
medium  bears  to  that  in  the  upper. 

That  is,  a3a(i  =  b,b3/n,  or  b2b3/a2at  =  n.  Therefore, 

sin.  i 

=  n 

sm.  r 

Since  the  velocity  of  light  through  a  non-crystalline  material 
such  as  glass  or  water  is  the  same  no  matter  what  the  direction 
of  the  rays  may  be,  and  depends  only  upon  the  nature  of  the 
material  and  the  wavelength  of  the  light,  the  above  equation 


INDEX  OF  *  REFRACTION  53 

indicates  that  the  ratio  of  the  sines  of  the  angles  of  incidence 
and  refraction  is  canst  ant,  that  is,  it  has  the  same  value  for 
all  different  angles  of  incidence.  This  statement,  known  as 
Snell's  law,  was  first  proved  by  direct  measurements  of  differ- 
ent sets  of  angles  of  incidence  and  refraction.  It.  holds  good  / 
for  all  isotropic  (non-crystalline)  materials,  but  not  —  as  we 
shall  see  later  —  for  all  crystals.  The  quantity  n,  which  was 
originally  defined  simply  as  the  ratio  of  the  sines  of  i  and  r, 
is  called  the  index  of  refraction  of  the  lower  medium;  in  tlier 
figure,  that  is  of  the  medium  into  which  the  light  is  refracted. 

The  index  differs  slightly  for  different  colors  or  wave-  / 
lengths.  It  is  for  this  reason  that  a  prism  not  only  bends  or 
refracts  a  beam  of  light,  but  also  separates  it  into  a  spectrum. 
Since  violet  is  bent  more  than  red,  evidently  the  index  is 
greater  for  the  shorter  waves  than  for  the  longer,  at  least  in 
ordinary  media  such  as  glass. 

We  have  assumed,  in  discussing  figure  23,  that  the  upper 
medium  is  the  free  ether,  but  the  conclusion  would  be  exactly 
the  same  if  it  were  any  other  isotropic  medium,  except  that 
the  corresponding  index  of  refraction  would  have  a  different 
value.  If  the  first  medium  were  air,  the  change  in  the  index 
would  be  extremely  slight,  since  light  travels  almost  as  fast 
through  air  as  through  a  vacuum.  But  if  it  were  water 
or  some  other  transparent  solid  or  liquid  the  change  would  be 
great.  In  such  a  case,  we  say  that  the  ratio  of  the  sines  —  or, 
what  comes  to  the  same  thing,  the  ratio  of  the  light  velocities  — 
is  the  index  of  the  second  medium  with  respect  to  the  first. 

Let  Ui  be  the  index  of  the  first  medium  (with  respect  to 
the  ether),  n2  that  of  the  second,  vx  the  velocity  of  light  in  the 
first  medium,  v2  that  in  the  second,  and  v0  the  velocity  in  the 
ether.  Then  * 


sin,  i  __  _y_1  ___  v^  _  VQ     .  v,_  _  ^  _^_  v^  _  n^ 
sin.  r       v2       v2v0       v2    x  V(i       v2    '   v,  "~  UL 

therefore,  if  n12  be  used  to  indicate  the  index  of  refraction  of 
the  second  medium  with  respect  to  the  first,  (light  passing  from 
the  first  to  the  second), 

n,2  =  n2/n1 


54  LIGHT 

Tabulated  values  of  the  refractive  indices  of  various  solids 
and  liquids  are  to  be  found  in  such  collections  of  physical  data 
as  the  Smithsonian  Physical  Tables,  Recueil  de  Constantes 
Physique  of  the  French  Physical  Society,  the  Physikalische- 
Chemische  Tabellen  of  Landolt  and  Btrnstein,  and  Kaye  and 
Laby's  Physical  Tables.  So  far  as  glass  is  concerned,  there  is 
an  almost  endless  variety  of  different  glasses,  having  all  sorts 
of  variations  of  refractive  index,  dispersion,  and  absorption. 

For  rough  calculations,  we  may  take  1.53  as  the  approxi- 
mate index  of  refraction  of  crown  glass  for  yellow  light,  and 
1.63  as  that  of  flint  glass.  The  index  of  water  for  yellow 
light  is  very  nearly  1.33,  that  of  diamond  2.42. 

It  is  customary  to  speak  of  substances  having  very  high 
refractive  indices  as  being  "optically  dense."  This  is  only  a 
technical  expression,  for  refractive  index  has  nothing  to  do 
with  real  density,  or  specific  gravity,  beyond  the  fact  that  in 
general  the  heavier  kinds  of  glass,  have  the  higher  indices. 
Thus,  we  say  that  diamond  is  an  extremely  dense  medium, 
and  that  flint  glass  is  denser  than  crown. 

25.  Total  reflection.  Critical  angle. — So  far,  we  have  taken 
up  principally  cases  where  the  light  passes  from  the  less  dense 
to  the  more  dense  medium,  as  from  air  to  water,  but  obviously 
cases  of  the  reverse  type  are  almost  as  common.  Whenever  we 
see  anything  that  lies  below  the  surface  of  water,  for  instance, 
the  light  must  pass  out  of  water  into  air  in  order  to  reach  our 
eyes.  In  such  a  case  we  may  still  write 

sin.  i 

— -—  n 

sin.  r 

where  i  means  the  true  angle  of  incidence  (on  the  water  side 
of  the  boundary),  r  the  true  angle  of  refraction  (on  the  air 
side),  and  n  is  the  index  of  air  with  respect  to  water,  just  the 
reciprocal  of  the  index  of  water  as  found  in  tables,  that  is, 
.75  instead  of  1.33.  Of  course,  beside  the  light  that  is  refracted 
out  into  the  air,  there  is  always  in  addition  a  reflected  beam 
going  back  into  the  water,  for  which  the  angle  of  reflection  is 
equal  to  the  angle  of  incidence,  exactly  as  it  would  be  if  the 
light  had  been  incident  on  the  air  side.  Sometimes  with  light 
incident  on  the  denser  side  of  a  dividing  surface,  it  will  hap- 
pen when  the  angle  of  incidence  is  large  enough,  that  there 


TOTAL  REFLECTION  55 

is  no  refracted  light  at  all,  all  instead  of  part  of  the  incident 
beam  being  reflected  back  into  the  first  medium.  A  considera- 
tion of  the  formula  of  refraction  shows  that  this  must  be  so. 
Solving  for  sin.  r,  we  get 

sin.  i 
sin.  r  = 


Since,  for  such  cases  as  we  are  now  considering,  n  is  less  than 
unity,  this  equation  shows  that  sin.  r  is  greater  than  sin.  i,  and 
therefore  r  is  greater  than  i,  both  being  acute  angles.  It  is 
possible,  then  for  r  to  be  equal  to  90°  and  sin.  r  equal  to  1, 
while  i  is  still  considerably  less  than  90°.  If  i  becomes  any 
greater,  sin.  r  as  calculated  from  the  above  equation  becomes 
greater  than  1 ;  and  this  means,  since  the  sine  of  a  real  angle* 
cannot  exceed  1,  that  there  is  no  refracted  wavefront.  Obvious- 
ly, the  largest  value  that  i  can  have  for  refraction  still  to 
occur,  is  that  value  which  makes  sin.  r  =  1.  Such  a  value  for 
the  angle  of  incidence  is  called  the  critical  angle.  If  we  let 
7  represent  the  critical  angle,  we  can  find  its  value  from  the 
equation  for  refraction,  by  substituting  7  for  -i,  and  1  for  sin.  r. 
This  gives 

sin.  7  =  n 

As  an  example,  let  us  calculate  the  critical  angle  for  crown 
glass,  in  contact  with  air.  We  have  taken  1.53  as  the  index 
from  air  to  the  glass,  which  gives  1/1.53,  or  .654  as  the  index 
from  the  glass  to  air.  Therefore 

sin.  7  =  .654 
7  =  40°49' 

So  much  for  the  mathematical  side  of  the  question.  The 
physical  interpretation  can  be  gotten  by  considering  figure  23 
again,  with  the  modification  that  now  the  velocity  of  light  in 
the  lower  medium  is  greater  than  that  in  the  upper.  Suppose, 
for  instance,  that  the  velocities  in  the  two  media,  and  the  angle 
of  incidence,  have  such  values  that  while  light  travels  the  dis- 
tance bobs  in  the  upper  medium  it  will  travel  a  distance  in  the 
lower  medium  greater  than  a2b3.  Under  these  circumstances 
the  radius  a2a.t  of  the  secondary  wavelet  from  a2  will  be  so 
great  that  the  point  b3  will  lie  ivithin  the  sphere  of  the  wave- 


56  LIGHT 

let  and  it  will  be  impossible  to  draw  a  plane  through  b3  tangent 
to  this  sphere.  When  the  angle  of  incidence  has  exactly  the 
critical  value,  a2a4  the  radius  of  the  secondary  wavelet  from 
a2,  is  just  equal  to  a2b3. 

One  of  the  most  effective  ways  of 
showing  total  reflection  is  that  illus- 
trated in  ^figure  24.  ABC  represents 
a  right-angled  prism  of  crown  glass. 
The  eye  is  held  at  some  such  point 
as  E,  close  to  one  of  the  shorter 
prism-faces.  SS'  is  any  rather  bright- 
ly illuminated  surface,  such  as  the 
sky,  the  whitewashed  wall  of  a  room, 
or  a  sheet  of  paper.  The  eye  sees, 
reflected  in  the  hypothenuse  BC,  an 

image  of  the  bright  area  SS',  the  upper  part  of  which  is 
almost  as  bright  as  SS'  itself,  the  lower  part  much  fainter. 
There  is  a  fairly  sharp  boundary  between  the  bright  upper 
part,  seen  by  total  reflection,  and  the  fainter  lower  part,  seen 
by  ordinary  partial  reflection.  Following  is  the  explanation. 

Any  point  of  SS',  such  as  a,  sends  out  rays  of  light  in  all 
directions,  but  only  a  small  bundle  of  these,  comprising  a  cone, 
will  reach  such  a  position,  after  reflection  at  the  face  BC 
and  refraction  at  the  faces  AB  and  AC,  that  they  can  enter 
the  pupil  of  the  eye  and  contribute  to  vision.  For  simplicity's 
sake,  a  single  ray  aa'a"E  is  drawn  to  represent  this  slender  cone. 
Similar  rays  are  drawn  from  a  few  other  points  on  SS'.  In 
every  case,  some  light  is  lost  by  reflection  at  the  two  surfaces 
AB  and  AC,  but  this  is  not  indicated  on  the  drawing  in  order 
that  the  diagram  may  not  become  too  intricate. 

There  will  be  some  point  on  SS',  such  as  b,  which  is  so 
situated  that  the  cone  of  light  from  it  which  enters  the  eye 
will  strike  the  hypothenuse  with  an  angle  of  incidence,  b'b"K, 
which  is  exactly  equal  to  the  critical  angle.  The  light  from 
any  point  to  the  left  of  b,  such  as  c  or  d,  will  strike  BC  at 
an  angle  less  than  the  critical  angle,  so  that  the  greater  part 
of  the)  light  in  the  small  cone  will  be  refracted  through  BC 
into  the  air,  below  and  to  the  right  of  the  prism,  leaving  only 
a  small  fraction  to  be  reflected  into  the  eye.  Therefore  such 


TOTAL  REFLECTION  57 

points  as  c  and  d  will  appear  faint  in  the  reflected  image.  On 
the  other  hand,  the  light  from  a,  or  any  other  point  to  the 
right  of  b,  will  be  reflected  from  BC  at  an  angle  greater  than 
the  critical  angle,  none  will  be  refracted  through  the  hypothe- 
nuse,  and  all  the  light  in  the  cone,  except  for  the  small  amount 
reflected  by  the  other  two  faces  of  the  prism,  will  enter  the 
eye.  Therefore,  points  to  the  right  of  b  will  appear  very 
bright  in  the  reflected  image  of  SS',  as  bright  as  if  they  had 
been  reflected  by  a  silvered  mirror,  or  even  brighter,  since  a 
silver  mirror  does  not  reflect  by  any  means  all  the  light  that 
falls  upon  it. 

Naturally,  the  critical  angle  for  any  material  depends 
not  only  upon  the  nature  of  the  material  itself,  but  also  upon 
that  of  the  material  in  contact  with  it.  For  example,  if  the 
prism  were  submerged  in  water  instead  of  being  in  air,  many 
of  the  rays  which  are  totally  reflected  against  air  would  be 
refracted  through  into  the  water. 

The  principle  of  the  totally  reflecting  prism  is  utilized  in 
a  number  of  optical  instruments,  a  few  of  which  we  shall  con- 
sider later.  But  one  of  the  most  interesting  cases  of  total 
reflection  is  seen  in  the  case  of  a  cut  diamond.  The  index  of  / 
diamond  with  respect  to  air  being  so  large,  its  critical  angle 
is  correspondingly  small,  and  a  diamond  owes  its  brilliance 
partly  to  this  fact  and  partly  to  the  additional  fact  that  its 
index  for  different  colors  differs  large- 
ly, so  that  for  certain  angles  of  in- 
cidence the  shorter  waves  are  totally 
reflected  while  the  longer  ones  are 
not.  Let  figure  25  represent  a  cross- 
section  of  a  cut  diamond.  Not  only 
the  ray  aaa,  but  the  very  oblique  one 
bbbb  may  undergo  total  reflection  at 
the  two  surfaces  XZ  and  YZ.  More- 
over, if  the  b  ray  happens  to  strike 

one  of  these  surfaces  at  the  critical  angle  for  green  light,  for  in- 
stance, the  waves  of  shorter  length  will  strike  it  at  an  angle 
greater  than  the  critical,  the  waves  of  greater  length  at  an 
angle  less.  For,  since  the  index  is  greater  for  short  than  for 
long  waves,  the  critical  angle  is  greater  for  long  than  for  short. 
Consequently,  red  rays,  will  escape  total  reflection  and  be  par- 


58 


LIGHT 


tially  refracted  out  of  the  diamond  for  angles  which  make  re- 
fraction impossible  for  violet  rays. 

Industrial  applications  of  the  total  reflection  principle  have 
been  made  in  the  manufacture  of  so-called  "shades"  flight- 
distributors)  for  incandescent  electric  lighting,  sidewalk-lights 
for  illuminating  basements,  etc. 

The  appearance,  to  a  fish,  of  things  outside  the  water,  is 
largely  affected  by  refraction,  "and  also  illustrates  total  re- 
flection. Figure  26  represents  a  pond,  the  eye  of  the  fish  being 

at  E.  XEY  is  the 
section  of  a  cone, 
whose  axis  is  verti- 
cal and  whose  half- 
angle  K  is  equal  to 
the  critical  angle  of 
water,  48°45'.  Any 
object  outside  the 
water,  as  at  a,  would 
be  seen  within  the 
Figure  26  con  e,  for  obviously 

light  passing  from  air  to  water  at  an  angle  of  incidence 
less  than  90°,  as  it  must  be,  will  be  refracted  with  an  angle 
of  refraction  which  is  less  than  what  would  be  the  criti- 
cal angle  for  light  incident  on  the  under  side  of  the 
surface.  The  object  a  would  appear  in  some  such  position 
as  a'.  It  is  clear  then  that  the  whole  array  of  objects 
outside  the  water  would  appear  to  the  fish  very  much  distorted 
from  their  true  relative  positions,  being  crowded  within  a 
relatively  small  cone  of  view.  Professor  R.  "W.  Wood,  of  Johns 
Hopkins  University,  has  taken  some  curious  photographs  which 
illustrate  what  he  calls  "fish-eye  views"  reproductions  of 
which  can  be  found  in  his  book  "Physical  Optics." 

Whatever  the  fish  sees  by  looking  at  the  surface  outside 
the  cone  XEY  would  be  totally  reflected  images  of  objects  in 
the  water.  For  instance,  it  would  see  the  objects  C  and  D, 
not  only  directly,  but  also  by  reflection  in  the  surface.  In  other 
words,  the  whole  top  surface,  outside  the  circle  whose  diameter 
is  XY,  would  appear  as  a  perfect  mirror.  Inside  this  circle, 
the  fish  would  see,  not  only  objects  that  are  outside  the  water, 
as  already  stated,  but  also  faint  reflections  of  objects  within 


DEVIATION  THROUGH  A  PRISM 


59 


the  water  between  M  and  N.  It  need  hardly  be  pointed  out 
that  as  the  fish  swims  about  the  cone  XEY  moves  with  it,  the 
diameter  XY  becoming  smaller  as  the  fish  approaches  the  sur- 
face, larger  as  it  sinks  toward  the  bottom. 

A  man  with  his  head  under  water  would  see  things  the 
same  way  as  a  fish  but  for  one  thing.  Our  eyes  are  adapted 
for  seeing  in  the  air,  and  the  index  of  refraction  of  the  cornea 
(the  forward  portion  of  the  eye)'  with  respect  to  air  is  such 
that  the  lens  within  the  eye  can  bring*  to  focus  upon  the  retina 
objects  anywhere  from  about  eight  inches  to  an  infinite  dis- 
tance away.  The  substitution  of  water  for  air,  as  the  medium 
in  contact  with  the  cornea,  alters  the  refraction  so  that  it  is 
impossible,  at  least  without  extreme  eye-strain,  to  focus  upon 
the  retina.  Therefore  human  vision  with  the  eyes  in  contact 
with  water  is  very  much  blurred  and  indistinct. 

26.  Deviation  through  a  prism. — We  shall  now  consider 
the  deviation  of  light  by  a  prism  (figure  27)  ;  and  since,  in 
the  actual  use  of  prisms  for  the  production  of  spectra,  the 
light  waves  are  practically  always  first  made  plane  by  the  use 
of  a  lens,  we  shall  take  only  the  case  of  plane  waves.  At 
each  surface  of  the  prism,  not  only  refraction  occurs,  but  also 
reflection;  but  in  this  discussion  we  shall  ignore  the  reflected 
light.  For  the  sake  of  symmetry, 
we  call  ix  the  angle  of  incidence 
at  the  first  surface,  rt  the  corre- 
sponding angle  of  refraction,  r2 
the  angle  of  incidence  at  the  sec- 
ond surface  and  i,  the  correspond- 
ing angle  of  refraction;  so  that 
i^  and  i,  are  angles  in  air,  rt  and 
r.,  angles  in  glass.  Then 


n  = 


sin.  i 


sin.  i2 


sm. 


sin.  r2 

where   n   is    the    index    of    glass 

with  respect  to  air.    The  drawing 

shows  a  series  of  wavefronts  sup- 

posed to  be  just  one  wavelength 

apart     (for    instance    the    lines    of    the    crests)     both    in    the 

air  and  in  the  glass,  though  of  course  the  actual  length  of  the 


v/ 

t 
Figure    27 


60  LIGHT 

light  waves,  as  compared  to  the  dimensions  of  a  practicable 
prism,  is  enormously  exaggerated  in  the  figure.  *It  will  be 
i  noticed  that  the  wavelength  is  shorter  in  glass  than  in  air) 
Indeed  this  must  be  so,  for  the  following  reasons :  The  period, 
or  time  of  one  vibration,  must  be  the  same  in  glass  as  in  air, 
for  only  so  many  waves  can  in  a  given  time  leave  thle  surface 
toward  the  glass  side  as  come  up  to  it  on  the  air  side.  Also, 
since  a  train  of  waves  advances  the  distance  of  one  wavelength 
during  the  time  of  one  vibration, 

wavelength 

velocity  — ^— — 

period 

Therefore,  since  the  period  is  the  same  in  the  two  media, 

wavelength  in  air          velocity  in  air 
wavelength  in  glass      velocity  in  glass 

Consequently,  whenever  light  of  any  wavelength  A  passes  from 
air  into  another  medium  whose  index    of    refraction   with    re- 
\  spect  to  air  is  n,  the  wavelength  within  this  medium  is  reduced 
to  the  value  A/n. 

The  angle  D,  between  the  wavefronts  of  the  light  before 
entering  the  prism  and  those  after  leaving  it,  or — what  comes 
to  the  same  thing — the  angle  between  the  rays  before  and  after 
passage  through  the  prism,  is  called  the  angle  of  deviation. 
The  refracting  angle  of  the  prism  itself  we  call  A.  It  can  be 
easily  proved  by  simple  geometry  from  the  figure  that 

A  — .  r,  +  r2 

0  =  1,4-12  —  A 

From  these  two  equations,  together  with  the  two  gotten  by 
applying  the  law  of  refraction  to  each  surface  of  the  prism, 
we  can,  if  A,  iit  and  n  are  given,  solve  for  r1?  i2,  r2,  and  D. 
A  and  n  are  necessarily  constant  for  a  given  prism  and  a  given 
wavelength  of  light,  but  by  turning  the  prism  about  an  axis 
perpendicular  to  the  plane  of  the  figure  i:  can  be  made  to  take 
any  value  from  0  to  90°.  Changing  the  value  of  it  in  such  a 
manner  will  naturally  cause  changes  in  the  value  of  D.  If  we 
plot  the  values  of  it  as  abscissae,  and  the  corresponding  values 
of  D  as  or  din  at  es,  the  curve  will  be  found  to  have  the  form 


MINIMUM  DEVIATION 


61 


shown  in  figure  28,  which  shows  that  for  a  certain  value  of  ix 
the  value  of  D  is  less  than  for  any  other  value  of  ix.    Both  ex- 
periment  and  theory — by   the         D 
application  of  the  differential 
calculus — show  that  this  mini- 
mum value  of  D  occurs  when  / 
ij  =  i2    and    rt  =  r2,    that    is 
when  the  light  passes  through 
the    prism   symmetrically.     In 
such  a  case, 

Figure   28 


n  = 


This  equation  has  a  great  deal  of  importance  in  practical 
optical  work.  For,  by  the  use  of  a  spectrometer,  both  the  re- 
fracting angle  of  a  prism  and  the  angle  of  minimum  devia- 
tion, D.  can  be  measured  with  great  accuracy.  Therefore,  by 
applying  this  equation  we  can  get  very  accurate  determinations 
of  the  index  of  refraction  for  any  piece  of  glass  that  can  be 
obtained  in  the  form  of  a  prism.  It  is  the  most  convenient 
method  for  finding  not  merely  the  average  index  for  white 
light,  but  the  separate  indices  for  different  wavelengths. 

Problems. 

1.  Calculate  the  angle  of  refraction    when    li^ht    strikes 
crown  glass  with  an  angle  of  incidence  of  60°. 

2.  Find  the  index  of  refraction  of  crown  glass  with  respect 
to  water,  for  yellow  light. 

3.  Light  within  a  piece  of  crown  glass  strikes  the  surface 
at  an  angle  of  incidence  of  40°.    At  what  angle  does  it  emerge? 


62  LIGHT 

4.  Calculate  the  critical  angle  for  diamond,  yellow  light. 

5.  Show  that  if  light  strikes  a  pile  of  parallel -sided  plates 
with  different  refractive  indices,  all  in  contact  with  one  another, 
it  enters  each  plate  with  the  same  angle  of  refraction  as  if  the 
others  were  absent,  and  finally  emerges  parallel  to  its  original 
direction. 

6.  A  star  seems  displaced  from  its  proper  position  owing 
to  refraction  in  the  earth's    atmosphere.     Show    that,    despite 
the  changes  in  the  density  of  the  air  at  different  levels,  we  can 
calculate  the  refraction    by    considering    that    all    the    air  has 
the  same  index  as  that  at  the  earth's  surface.     (See  problem 
5). 

7.  Find  the  approximate  length  of  wave,  in  water,  of  the 
extreme  red  and  the  extreme  violet  light,  assuming  the  index 
for  both  to  be  1.33. 

8.  Calculate   the   angle  of  minimum   deviation   for  a   60° 
prism,  the  light  having  index  1.68. 

9.  What  must  be  the  properties  of  a  body  which,  in  air, 
would   be   invisible  under   any  illumination?     Would   such   a 
body  be  visible  if  immersed  in  water  f 

10.  Certain  aquatic  bodies  are  "nearly  invisible  in  water. 
What  are  their  properties'? 

11.  Show  that  any  colorless  and  transparent  object  would 
be  invisible  if  surrounded  completely  by  uniformly  illuminated 
walls. 

12.  Prove  that,  in  figure  27,  A  =  rt  +  r,  and  D  =  it  +  i2 
—  A. 

13.  A  real  diamond  will  continue  to  glitter  when  immersed 
in  water,  while  an  imitation  will  not.     Explain  this. 

14.  Plot  four  points  on  a  curve  like  figure  28,  for  a  prism 
of  60°,  having  an  index  1.54. 


CHAPTER  V. 

27.  Reflection  and  refraction  of  spherical  waves  at  a  plane  surface. 
— 28.  Judgment  of  the  distance  of  an  image. — 29.  Image  of  an  extended 
object. — 30.  Reflection  and  refraction  at  spherical  surfaces. — 31.  Lenses. 
— 32.  Two  lenses  in  contact. — 33.  Chromatic  aberration. — 34.  Achro- 
matic lenses. — 35.  Image  of  an  extended  object.  Undeviated  ray. — 36. 
Magnification. — 37.  Micrometer.— 38.  Imperfections  in  mirrors  and 
lenses. — 39.  Spherical  aberration. — 40.  Curvature  of  field. — 41.  Astig- 
matism.— 42.  Lenses  for  special  purposes. 

27.  Reflection  and  refraction  of  spherical  waves  at  a  plane 
surface. — It  is  shown  in  the  preceding  chapter  that  when  plane 
waves  strike  a  plane  surface,  both  the  refracted  and  the  re- 
flected wavefronts  are  plane.  In  this  chapter  we  shall  show 
that  when  the  incident  wavefronts  are  spherical  (diverging^ 
from  a  point)  and  the  reflecting  surface  plane,  the  reflected 
wavefronts  are  also  spherical  but  the  refracted  wavefronts  are 
not,  except  as  an  approximation  to  the  truth.  We  shall  also 
show  that  when  both  the  incident  wavefronts  and  the  reflecting 
surface  are  spherical,  neither  the  reflected  nor  the  refracted 
wavefronts  are  truly  spherical,  except  in  the  one  special  case 
that  the  center  of  the  incident  waves  coincides  with  that  of  the 
reflecting  surface. 

In  figure  29,  let  CBD 
represent  the  plane  boundary 
between  two  media,  the  in- 
dex of  refraction  of  the  low- 
er with  respect  to  the  upper 
being  n.  A  is  the  center  of 
a  system  of  spherical  wave- 
fronts,  advancing  from  A 
toward  the  surface  CBD.  B 
is  the  foot  of  the  perpendic- 
ular from  A  upon  this  sur- 
face, that  is,  the  first  point 

on    CBD    reached    by    each  Figure  29 

wavefront.  If  the  reflecting  surface  had  not  been  in  its  place,  a 
wavefront,  after  reaching  B  would  continue  to  travel  with  its 
original  velocity  and  in  a  short  time  would  reach  some  such 

(63) 


64  LIGHT 

position  as  CxD.  With  the  surface  in  its  place,  B  becomes  the 
center  for  a  secondary  wavelet  in  the  upper  medium  whose 
radius,  at  the  instant  when  the  incident  wavefront  reaches  C 
and  D,  will  have  acquired  the  length  Bx'  =  Bx.  In  the  mean- 
time, other  points  such  as  M  and  N  will  also  have  been  reached 
by  the  incident  wave  and  become  centers  of  secondary  wave- 
lets, whose  radii  in  the  upper  medium  will  be  respectively 
MO'  =  MO  and  NP'  =  NP.  The  reflected  wavefront  will 
therefore  be  CO'x'P'D,  the  envelope  of  all  such  secondary 
wavelets.  From  the  manner  of  its  formation,  it  is  obviously 
exactly  symmetrical  with  the  hypothetical  incident  wavefront 
COxPD,  and  therefore  is  truly  spherical,  with  center  at  A'.  A 
and  A'  are  equally  distant  from  the  reflecting  surface,  and  the 
line  AA'  is  perpendicular  to  the  latter.  An  eye  placed  any- 
where in  the  upper  medium  would  receive  reflected  light  which 
would  appear  to  come  from  A',  though  really  from  A,  and  we 
.  therefore  say  that  A7  is  the  reflected  image,  or  image  by  Te-\ 
flection,  of  A. 

The  refracted  wavefront  is  formed  in  a  simitar  manner, 
except  that  in  the  lower  medium  the  radii  of  the  secondary 
wavelets  are  shorter  than  in  the  upper  if  n  is  greater  than 
unity,  longer  if  n  is  less  than  unity.  The  radius  By  of  the 
secondary  wavelet  from  B  as  center  is  not  equal  to  Bx,  but  to 
Bx/n.  The  radius  of  the  secondary  wavelet  from  M  is  Mu  =: 
MO/n,  that  from  N'is  Nv  =  NP/n,  etc.  The  envelope  of  all 
these  secondary  wavelets  in  the  lower  iBedium,  CuyvD,  turns 
out  to  be,  not  a  sphere,  but  a  surface  of  higher  order.  There- 
fore, we  cannot  say  that  there  is  a  refracted  image  in  the  same 
strict  sense  in  which  we  speak  of  a  reflected  image.  It  is  true 
*  that  an  eye  placed  in  the  lower  medium  would  receive  light 
that  appeared  to  come  from  some  point  in  the  upper  medium 
other  than  the  true  source  A,  but  this  apparent  "image"  would 
have  a  different  position  for  every  change  in  the  location  of 
the  eye. 

However,  it  is  always  possible  to  describe  a  sphere  which 
approximates  more  or  less  closely  to  the  refracted  wavefront. 
Suppose,  for  example,  that  a  circular  arc  be  drawn  through 
the  three  points  C,  y,  and  D,  and  a  spherical  segment  be  formed 
by  rotating  this  arc  about  the  axis  AA'.  This  surface  would 


SPHERICAL  WAVES  AT  A  PLANE  SURFACE       65 

coincide  exactly  with  the  refracted  wavefront  at  the  three 
given  points,  and  would  also  pass  very  close  to  other  points 
such  as  u  and  v.  The  nearer  the  three  points  C,  y,  and  D  are 
together,  the  closer  would  sphere  and  wavefront  coincide  for 
all  the  region  between  C  and  D,  and  if  the  three  points  are 
distant  from  one  another  by  only  an  infinitesimal  amount  we 
may  say  that  in  the  immediate  neighborhood  of  these  points 
the  coincidence  is  exact.  If  A"  represents  the  center  of  this 
sphere,  then  to  an  eye  located  in  the  lower  medium  anywhere 
along  the  line  AA'  or  AA'  produced,  or  in  the  close  neighbor- 
hood of  this  line,  as  at  the  point  E,  the  light  would  appear  to 
come  from  A"  instead  of  from  A,  and  we  therefore  define  A" 
as  the  image  by  refraction  of  A.  But,  if  the  eye  be  located  at 
some  distance  from  the  line  AA',  as  at  E',  the  light  appears  to 
come  from  a  different  point,  such  as  Q. 

In  order  to  find  the  position  of  the  image  A",  it  is  best 
to  consider  the  refracted  wavefront  just  as  it  breaks  through 
the  surface  CBD,  that  is,  we  imagine  C  and  D  to  be  very  close 
together  and  By  and  Bx  to  be  infinitesimally  small.  Then  the 
radius  of  the  sphere  through  C,  y,  and  D  will  be  the  radius  of 
the  refracted  wavefront  just  as  it  breaks  through  the  surface, 
and  equal  to  the  distance  of  the  image  A"  from  the  surface. 

To  find  A"  is  merely  a  matter  of 
plane  geometry.  CD  is  a  chord  common 
to  the  two  arcs  CxD  (hypothetical  inci- 
dent wavefront)  and  CyD  (refracted 
wavefront).  The  line  drawn  from  the 
middle  point  of  a  chord,  perpendicular  to 
the  latter,  till  it  meets  the  arc,  is  called 
the  sagitta  of  the  arc.  Thus,  Bx  is  the 
sagitta  of  the  arc  of  the  incident  wave- 
front,  By  that  of  the  refracted  wave- 
front.  We  must  first  find  what  relation  the  sagitta  bears  to 
the  radius.  This  can  be  best  done  by  a  consideration  of  figure 
30,  where  the  complete  circle  is  shown.  The  triangles  KBD 
and  DBy  are  similar,  therefore 

KB    _BD 
BD   "BY 

If  we  let  s  represent  the  sagitta,  R  the  radius  of  the  circle,  and 


66  LIGHT 

a  the  half -chord,  KB  =  2R  —  s,  BD  =  0,  and  By  =  s.     There- 
fore 


2R  — s 


We  are  supposing  that  C,  y  and  D  are  very  close  together,  so 
that  both  s  and  a  are  quantities  of  infinitesimal  magnitude, 
but  R  remains  a  quantity  of  finite  size,  therefore  in  the  limit 
2R  —  s  becomes  equal  to  2R,  and 

JL       !_ 
a2  "  2R 


or, 


E= 


Equation  (1)  shows  that  although  a  and  s  both  become  in- 
finitesimally  small,  the  ratio  o2/s  remains  a  finite  quantity, 
equal  to  the  diameter  of  the  circle. 

Now  let  us  apply  equation  (1)  to  both  the  hypothetical 
incident  wavefront  and  the  refracted  wavefront.  For  the 
former,  s=Bx,  R=BA.  For  the  latter,  s=By,  R=BA",  the 
required  distance.  Therefore,  since  a  is  the  same  for  both, 
viz.,  BD, 


If  we  divide  the  last  equation  by  the  one  above  it,  we  get 

B A"  _  Bx 
BA  ~~  By 

But,  by  the  method  of  constructing   the  refracted  wavefront, 
Bx/By  .-  n.    Therefore, 

BA" 
•^-r-  =n 


JUDGMENT   OF  DISTANCE  67 

or  in  general,  if  dx  represent  the  distance  of  any  point  source 
of  light  from  a  plane  refracting  surface,  d2  the  distance  of  the 
corresponding  refracted  image,  and  n  the  index  of  refraction 
of  the  second  medium  with  respect  to  the  first,  then. 


As  an  application  of  formula  (2)  suppose  a  certain  object 
to  be  2  feet  above  the  surface  of  a  pond.  A  person  above  the 
surface  would  of  course  see  a  reflected  image  of  it,  apparently 
2  feet  beneath  the  surface,  but  a  fish  in  the  water  would  see 
a  refracted  image.  If  the  fish  is  directly  beneath  the  object, 
we  can  apply  equation  2,  putting  d  =  2,  n  =  %,  or  1.33.  This 
gives 

d2  =  2  X  ^r-~  2.67  feet 
o 

Tha-t  is,  the  object  would  appear  to  the  fish  to  be  2  ft.  8  in. 
above  the  surface. 

If  the  source  of  light  lies  within  the  denser  medium,  the 
refracted  light  travels  faster  than  the  incident,  the  refracted 
wavefront  of  figure  29  will  be  bulged  out  more  than  the  inci- 
dent instead  of  being  flattened,  and  n  has  a  value  less  tihan 
unity.  Thus,  suppose  we  look  straight  down  to  the  bottom  of 
a  pool  2  feet  deep.  Then,  in  equation  (2),  dt  =  2,  n  =  % ..  and 

d2  =  2Xj  =1-5  feet 
4 

The  pool  appears  to  be  only  %  its  actual  depth.  The  fact  that 
the  wavefronts  are  not  spherical  is  shown  clearly  by  the 
observation  that,  if  we  look  very  obliquely  to  the  bottom,  the 
depth  appears  to  have  much  less  than  %  its  actual  value.  This 
explains  a  curious  phenomenon  which  anyone  standing  in  a 
pool  a  few  feet  deep  with  perfectly  level  bottom  can  hardly 
fail  to  notice.  The  bottom  always  appears  to  be  bowl-shaped, 
with  the  greatest  depth  just  underfoot,  and  that  depth  of 
course  just  about  %  the  true  depth  of  the  whole  pool. 

28.  Judgment  of  the  distance  of  an  image. — The  student 
may  wonder  why  the  curvature  of  the  wavefront  has  anything 
to  do  with  our  judgment  of  the  distance  of  an  object  perceived, 


\ 


68  LIGHT 

for  one  is  apt  to  think  that  only  a  single  ray  ever  enters  the  eye. 
This  is  incorrect,  for  the  pupil  of  the  eye  has  a  finite  size  and 
therefore  always  receives  a  finite  area  of  the  wavefront.  Ac- 
cording as  the  curvature  of  this  section  of  wavefront  is  greater 
or  less,  we  must  exert  more  or  less  muscular  strain  upon  the 
lens  in  the  eye  in  order)  to  focus)  the  light  upon  the  retina,  and 
the  degree  of  this  strain  enables  us  to  judge  distance  to  some 
extent.  Still  more  important  is1  the  fact  that  we  ordinarily  see 
with  both  e^yes  at  once,  thus  taking  in  at  the  same  time  two 
separate  sections  of  wavefront.  A  person  with  only  one  eye 
is  far  less  accurate  in  estimating  distance  than  a  normal  per- 
son. For  instance,  a  one-eyed  man  usually  has  much  greater 
difficult}7-  in  hitting  a  nail  with  a  hammer  for  this  reason. 

Naturally  enough,  estimation  of  distance  becomes  much 
more  difficult,  even  for  normal  two-eyed  vision,  when  the  dis- 
tance becomes  great.  It  is  comparatively  easy  to  tell  whether 
an  object  is  five  feet  away  or  ten  feet,  for  the  difference  in 
curvature  of  a  five  foot  and  a  ten  foot  sphere  is  comparatively 
great.  But  it  is  not  easy  to  tell  whether  a  distant  object  is, 
nearer  to  a  mile  or  to  two  miles  from  us.  For  a  small  section 
of  a  sphere  of  either  one  or  two  miles  radius  is  nearly  flat,  and 
there  is  no  perceptible  difference  in  the  focussing  and  align- 
ment of  the  eyes  for  such  great  distances.  Our  estimation  of 
great  distances  is  a  result  of  subconscious  consideration  of  such 
details  as  size,  speed  of  motion  (if  the  object  seen  happens  to- 
be  in  motion),  distinctness  of  vision,  etc.,  and  at  best  it  is  very 
uncertain  and  subject  to  queer  illusions.  Soldiers  are  given  a 
long  and  systematic  training  in  judging  distance. 

If  our  eyes  were  set  three  feet  apart  instead  of  a  few 
inches,  judgment  of  distance  would  become  much  easier.  No 
doubt  small  animals  with  their  eyes  close  together  are  less 
adept  in  this  respect  than  human  beings.  Indeed  there  is  some 
reason  fori  thinking  that  many  animals  and  birds  are  much 
less  keen  than  men  in  observing  stationary  objects,  although  a 
moving  objects  almost  instantly  arrests  their  attention.  Most 
birds  have  the  eyes  set  in  the  sides  of  the  head,  and  therefore 
seie  an  object  with  only  ones  eye  at  a  time.  This  fact  must 
seriously  hinder  them  in  the  estimation  of  distance;  and  it 
is  no  doubt  to  counteract  this  deficiency  that  birds  have  the 


IMAGE  OF  AN  EXTENDED  OBJECT  69 

habit,  particularly  when  alarmed,  of  darting  the  head  rapidly 
forward  and  backward,  so  as  to  get  two  or  more  points  of  view 
of  any  object  that  excites  their  suspicion.  The  parallax  of  the 
observed  object  gives  some  ground  for  a  judgment  of  its  remote- 
ness. Thus  the  mechanism  by  which  a  bird  estimates  distance 
is  perhaps  an  automatic  and  unconscious  application  of  the 
same  principle  used  by  a  surveyor  in  finding  the  width  of  a 
river,  or  by  an  astronomer  in  finding  the  distance  of  the  nearer 
fixed  stars. 

29.  Image  of  an  extended  object. — We  have  so  far  con- 
sidered only  cases  where  a  single  point  acted  as  a  source  of 
light,  and  have  found  the  positions  of  the  reflected  image  and 
the  refracted  image  of  this  point  source.    Actually,  we  always 
have  to  deal  with  objects  more  or  less  extended;  but  in  order 
to  find  the  images  of  such  an  object  we 

have  only  to  apply  the  principles  al- 
ready learned  to  each  point  of  it.  In 
figure  31,  AB  represents  any  object. 
The  point  A  has  an  image  by  reflection,  _ 
A',  and  an  image  by  refraction,  A", 
found  by  these  principles,  and  similar- 
ly B  has  an  image  by  reflection,  B',  and 
an  image  by  refraction  B",  and  so  on.  Figure  31 

So  far  as  the  image  by  reflection  goes,  it  is  exactly  symmetri- 
cal to  the  object,  with  respect  to  the  reflecting  plane. 

30.  Reflection  and  refraction  at  spherical  surfaces. — We 
shall  now  consider  reflection  and  refraction  at  spherical  sur- 
faces, a  subject  which  is  of  great  importance  because  of  the 
use  of  curved  mirrors  and  lenses  in  optical  instruments. 

Figures  32  to  35  show  four  different  cases  of  the  reflection 
and  refraction  of  spherical  wavefronts  at  a  spherical  surface, 
the  source  of  the  incident  waves  being  in  each  case  on  the  con- 
cave side.  All  these  figures  are  drawn  on  the  supposition  that 
the  medium  on  the  convex  side  (second  medium),  is  denser 
than  that  on  the  concave  side  (first  medium),  and  both  media 
are  supposed  to  extend  indefinitely.  The  reflecting  and  re- 
fracting surface  is  indicated  by  a  heavy  line,  the  incident  wave- 
fronts  by  normal  unbroken  lines,  and  the  reflected  and  refracted 
wavefronts  by  dotted  lines.  The  two  last  are  not  truly  spheri- 
cal, but  are  near  enough  to  be  considered  so  as  long  as  the 
Incidence  is  nowhere  very  oblique.  Therefore,  the  center  of 


70 


LIGHT 


the  incident  wavefronts  is  represented  by  0,  the  center  of  the 
reflected  wavefronts  (except  in  figure  35)  by  I,  the  center  of 
the  refracted  wavefronts  by  I',  and  the  center  of  the  surface 
itself  by  C.  Arrows  show  the  direction  of  advance  of  the 
waves. 


Figure  33 


t'o 


,-v^r 


Figure   34  Figure  35 

In  figure  "B3,  the  source  is  farther  from  the  mirror  than  is 
the  center  C.  The  reflected  waves  converge  to  the  point  I, 
betAveen  C  and  the  mirror,  but  nearer  to  the  former,  and 
diverge  again  after*  passing  through  I.  The  refracted  waves 
diverge  more  than  the  incident,  appearing  therefore  to  come 
from  a  point  I',  between  C  and  0. 

In  figure  &2i  -the  source  0  is  between  C  and  the  mirror,' 
nearer  to  the  former.  The  point  I,  to  which  the  reflected  waves 
converge,  and  from  which'  they  later  diverge,  lies  outside  of  C. 
The  refracted  waves  diverge  less  than  the  incident,  appearing 
to  come  from  I'  between  0  and  C. 

^  In  figure  34  0  is  nearer  to  the  surface  than  to  C.  Here 
the  reflected  waves  do  not  converge  at  all,  but  diverge  at  once, 
seeming  to  come  from  the  point  I  within  the  second  medium. 


SPHERICAL  SURFACES 


71 


The  center  of  the  refracted  waves  is  again  between  0  and  C. 

In  figure  35  0  is  just  halfway  between  C  and  the  surface. 
The  reflected  waves  are  plane,  and  may  be  said  either  to 
converge  to  an  infinitely  distant  point  on  the  left,  or  to  diverge 
from  an  infinitely  distant  point  on  the  right.  The  latter  state- 
ment is  preferable,  since  an  eye  placed  in  the  path  of  the  re- 
flected light  would  see  an  image  of  0  infinitely  distant  on  the 
right  of  the  figure.  In  fact  these  waves  would  strike  the  eye 
exactly  as  waves  coming  from  a  very  distant  star. 

The  points  I  in  figures  32  and  33  are  said  to  be  real  images 
of  the  corresponding'  sources  0,  because  the  light  is  actually 
converged  to  these  points;  while  in  the  other  two  figures  I  is 
only  a  virtual  image,  the  reflected  light  not  actually  being 
brought  to.'  focus  at  these  points,  but  simply  diverging  as  if 
it  had  come  from  them.  In  each  of  the  figures,  the  refracted 
image,  I'  is  virtual. 


Figure  36 


Figure   37 


?.  I 


<x.  .... 

?>Af'\  \U1 


Figure   38  Figure  39 

Figures  36  to  39  differ  from  the  four  preceding  figures  in 
that  the  reflecting  and  refracting  surface  is  convex,  instead  of 
concave,  to  the  incident  light  and  the  rarer  medium.  In  each 


72  LIGHT 

of  these  cases,  the  reflected  rays  diverge  from  the  surface  at 
once,  and  I  is  therefore  a  virtual  image.  As  shown  in  figure 
38,  there  is  a  certain  position  for  the  source  0,  depending  upon 
the  radius  of  the  surface  and  the  index  of  refraction,  for  which 
the  refracted  waves  are  plane.  If  0  lies  any  nearer  to  the 
surface,  as  in  figure  39,  they  diverge,  and  V  is  virtual.  If  O 
is  farther  from  the  surface,  they  converge,  and  I'  is  real,  as 
in  figures  36  and  37.  In  figure  36  the  source  is  infinitely 
distant,  and  the  incident  waves  are  plane.* 

From  what  precedes,  it  is  obvious  that  there  is  a  great 
diversity  of  typical  cases  for  reflection  and  refraction  at 
spherical  surfaces.  The  surface  may  be  concave  or  convex  to 
the  incident  light:  the  source  may  lie  in  the  medium  of  less 
or  of  greater  optical  density,  and  may  be  at  any  distance  from 
the  surface.  To  develop  and  remember  for  each  case  a  special 
formula  giving!  the  positions  of  I  and  V  in  terms  of  that  of  0, 
would  be  unduly  laborious.  Fortunately,  we  can  derive  a  pair 
of  very  general  formulae,  which  are  applicable  to  any  case 
that  may  arise,  provided  we  make  consistent  and  rational  con- 
ventions in  regard  to  the  algebraic  sign  of  the  distances  in- 
volved. 

We  shall  suppose  all  distances  to  be  measured  from  the 
reflecting  surface  as  a  base,  distances  to  one  side  being  regarded 
as  positive,  those  to  the  other  side  negative;  and  it  will  be 
most  convenient  to  take  the  side  from  which  the  light  comes 
as  the  positive  side.  We  let  r  stand  for  the  radius  of  the  re- 
flecting and  refracting  surface,  u  for  that  of  the  incident 
wavefronts,  v  for  that  of  the  reflected  wavefronts,  and  v'  for 
that  of  the  refracted  wavefronts.  In  figures  32  and  33  all 
these  quantities  are  positive.  In  figure  34  v  has  become 
negative,  while  it  is  ±00  in  figure  35.  v'  is  negative  in  figures 
36  and  37,  -+-_  oo  in  figure  38,  and  positive  in  all  the  other 
cases  shown,  r  is  always  positive  for  a  concave  mirror,  nega- 
tive for  a  convex  one.  u  is  always  and  necessarily  positive, 
unless  the  incident  waves  are  rendered  convergent  before 
striking  the  surface,  by  means  of  another  mirror  or  a  lens. 

*In  figures  32  to  35  the  index  of  refraction  has  been  taken  as  1.5, 
but  in  figures  36  to  39  it  has  been  taken  as  1.67  to  avoid  making  some 
of  these  figures  inconveniently  long. 


SPHERICAL  SURFACES 


73 


It  is  easily  seen  that  v  is  positive  when  the  reflected  image 
is  real,  negative  when  it  is  virtual.  On  the  other  hand,  v'  is 
negative  when  the  refracted  image  is  real,  positive  when  it  is 
virtual. 

It  is  easiest  to  derive  the  two  formulae  by  considering  a 
case  where  all  the  quantities,  r,  u,  v,  and  v'  are  positive,  as 
in  figure  32.  Figure  40  represents  the  case  of  figure  32  some- 


J?_-_-L-9- 


Figure    40 

what  exaggerated  to  make  the  diagram  clearer.  I,  C,  If  and 
0  have  the  same  significance  as  in  the  preceding  figures.  XPY 
is  the  reflecting  and  refracting  surface  (center  at  C),  AdB 
an  incident  wavefront  as  it  would  be  if  it  advanced  into  the 
second  medium  without  being  retarded  (center  at  0),  AfB 
the  actual  refracted  wavefront  (center  at  I'),  and  AeB  the 
reflected  wavefront  (center  at  I).  In  practice,  mirrors  are 
seldom  used  in  which  the  diameter  XY  of  the  mirror-faee,  is 
more  than  %  of  the  radius  of  curvature  CP  =  r.  In  this 
figure  XY  is  made  about  equal  to  r  in  order  that  the  different 
arcs  having  a  common  chord  AB  may  be  more  clearly  seen  as 
separate. 

The  formula  for  the  reflected  wave  is  based  upon  the  fol- 
lowing physical  fact:  While  the  incident  light  would,  but  for 
retardation  in  the  second  medium,  travel  from  P  to  d,  the  re- 
flected light  travels  back  from  P  to  e.  Therefore  the  distances 
Pd  and  Pe  are  equal.  Putting  this  statement  into  the  form  of 
an  equation, 

Pe  =  Pd  (3) 


74  LIGHT 

But  we  can  write 

Pe=:PK  —  eK 

Pd=dK  —  PK 
Therefore 

PK  —  eK  =  dK  —  PK 

dK  +  eK  =  2PK 

But  dK,  eK,  and  PK  are  respectively  the  sagittas  of  the  in- 
cident wavefront,  the  reflected  wavefront,  and  the  reflecting 
surface.  We  can  therefore  apply  to  each  of  them  the  general 
geometrical  formula  (1),  getting 

dK  =  a2/2u 
eK  —  a2/2v 
PK  =  a2/2r 
Making  these  substitutions,  we  get 

_o^         _a?_  _      2a^ 

2u  "h  2v  "      2r 
or,       ' 


This  is  the  general  formula  for  a  mirror.  The  symbol  f, 
equal  to  r/2,  is  called  the  focal  length  of  the  mirror.  Its 
physical  meaning  can  be  shown  by  supposing  that  u  =  oo  ,  that 
is,  that  we  are  dealing  with  an  object  infinitely  distant.  Then 
1/u  =  0,  1/v  =  1/f  ,  and  v  —  f  .  Then  f  is  the  distance  from  the 
mirror  to  that  point  (called  the  principal  focus)  where  parallel 
rays,  or  plane  waves  are  brought  to  a  focus  in  the  reflected  light. 
Conversely,  if  u  =  f  ,  1/v  =  0,  and  v  =  oo  ;  that  is,  if  the 
source  of  light  is  at  the  principal  focus,  the  reflected  waves 
are  plane.  If  neither  u  nor  v  is  infinite,  the  points  O  and  I 
are  called  conjugate  foci.  If  the  source  is  at  0,  the  reflected 
image  is  at  I,  and  conversely  if  the  source  is  at  I  the  image 
will  be  at  0,  for  equation  (4)  shows  that  the  relation  between 
these  two  points  is  reciprocal,  u  and  v  appearing  in  it  in  ex- 
actly the  same  way. 

The  formula  for  the  refracted  wave  is  found  in  a  similar 
way.     The  physical  fact  upon  which  it  is  based  is  this  :   While 


SPHERICAL  SURFACES  75 

the  incident  light  would,  but  for  the  retardation  in  the  second 
medium,  travel  from  P  to  d,  the  actual  refracted  wave  travels 
only  from  P  to  f,  where  Pd  and  Pf  are  to  each  other  as  the 
velocities  of  light  in  the  two  media.  That  is, 

Pd 

FF=n 

or 

Pd  =  n  X  Pf 
But 

Pd=:dK  —  PK 
Pf  ^:fK  —  PK 

(5) 
therefore 

dK  —  PK=rn(fK  —  PK) 

nXfK  —  dK=  (n  —  1)PK 

But  fK,  dK,  and  PK  are  the  sagittas  respectively  of  the  re- 
fracted wavefront,  the  incident  wavefront,  and  the  refracting 
surface.  Applying  to  each  of  these  the  geometrical  formula 
(1),  we  get 

f  K  =  a2/2v' 

dK  =  a2/2u 
'PK  =  a2/2r 
With  these  substitutions, 

nXa2    _**  _  (n  —  l)a2 

2v'      "  2u  ~  2r 

or 


This  formula,  for  the  refracted  light,  is  necessarily  more  com- 
plicated than  that  for  the  reflected,  because  it  involves  the 
index  of  refraction,  which  does  not  affect  reflection,  and  there- 
fore does  not  appear  in  (4). 

Formula  (6)  is  of  less  common  use  than  (4).  but  there 
are  certain  problems  where  it  becomes  necessary,  for  example, 
the  following. 


76  LIGHT 

A  spherical  globe,  one  meter  in  diameter,  and  made  of 
thin  glass,  is  filled  with  water.  A  small  fish  is  located  40cm. 
from  the  glass  wall  at  a  certain  side  of  the  globe,  (a)  Where 
would  be  the  image  which  the  fish  would  see  of  himself  in  the 
farthest  part  of  the  surface!  (b)  Where  would  the  fish  appear, 
to  a  person  outside  the  globe  on  the  side  farthest  from  the  fish? 

Question  (a)  is  easily  answered,  for  the  fish  would  see  his 
own  reflected  image,  and  we  only  need  to  apply  equation  (4) 
as  if  the  glass  wall  were  non-existent,  since  its  thinness  pre- 
vents it  from  affecting  the  problem  to  any  extent.  Then  we 
must  put  r  =  50,  u  =  60,  giving  v  =  42.9cm.  Therefore  the 
fish  would  see,  reflected  from  the  farthest  part  of  the  boundary 
of  the  globe,  an  image  of  himself  42.9cm.  from  the  boundary, 
17.1  cm.  from  himself.  (If  we  had  tried  to  find  the  location 
of  the  image  of  the  fish  formed  by  reflection  in  the  nearest 
part  of  the  wall,  it  would  have  come  out  to  be  behind  the  fish 
and  therefore  not  discernable  by  him  as  an  image). 

To  answer  question  (b),  we  must  apply  equation  (6),  for 
we  have  to  do  with  refracted  light.  Since  the  light  passes 
from  water  to  air,  the  appropriate  index  of  refraction  is  not 
%,  but  the  reciprocal  of  this,  %.  Therefore 

31  1 


4v'       60-    "4X50 

Giving  v'  =  64.3.  Therefore  the  person  outside  the  globe  would 
see  the  fish  apparently  4.3  cm  farther  away  than  it  really  is. 
The  fact  that  v'  comes  out  positive  shows  that  the  refracted 
image  is  on  the  same  side  of  the  bounding  surface  as  the  fish 
itself.  The  reflected  image  seen  by  the  fish  is  real,  the  re- 
fracted image  seen  by  the  observer  outside  is  virtual. 

Mirrors  for  experimental  work  in  optics  are  usually  either 
flat  or  concave,  though  convex  mirrors  are  occasionally  used. 
They  are  made  by  taking  a  disc  of  homogeneous  and  thoroughly 
annealed  glass,  and  reducing  one  surface  to  the  required  radius 
of  curvature  by  careful  grinding  and  polishing.  This  surface 
is  then  covered  with  a  deposit  of  silver  by  chemical  deposition 
from  a  solution  of  silver  nitrate,  and  the  silver  film  is  thorough- 
ly dried  and  then  lightly  polished  with  chamois  and  rouge. 
The  silver  of  course  prevents  any  appreciable  refracted  light 
so  that  the  major  part  of  the  incident  light  is  turned  into  the 


LtfNSES 


77 


reflected  beam.  Flat  mirrors  such  as  were  used  in  the  experi- 
ments of  Fizeau  and  Foucault  for  finding  the  velocity  of  light, 
whose  function  is  to  reflect  only  part  of  the  light  and  transmit 
the  rest,  must  of  course  be  ground  and  polished  flat  on  both, 
sides.  One  face  is  then  covered  with  a  very  thin  film  of  silver, 
the  best  result  being  obtained  when  the  film  reflects  approxi- 
mately half  the  incident  light,  transmitting  the  rest,  except 
for  some  unavoidable  absorption.  Such  a  mirror  is  said  to  be 
half -silvered.  When  a  silvered  mirror  becomes  tarnished  and 
dull,  it  is  a  comparatively  easy  matter  to  dissolve  off  the  old 
silver  with  nitric  acid  and  put  a  new  silver  coating  upon  it. 

31.  Lenses. — A  lens  is  a  disc  of  transparent  refractive  ma- 
terial, such  as  glass,  bounded  by  two  surfaces,  one  or  both  of 
which  is  curved,  usually  spherical.  Each  surface  produces 
some  reflection,  which  not  only  weakens  the  transmitted  beam 
but  also  causes  annoyance  in  other  ways.  The  light  reflected 
from  the  lens-surfaces  is  therefore  a  hindrance,  but  is  abso- 
lutely unavoidable.  In  the  following  discussion  of  lenses  we 
shall  ignore  the  reflected  light. 


Figure    41 

When  spherical  wavefronts,  with  center  at  0  (figure  41) 
strike  a  lens,  they  are  refracted  at  the  first  surface,  and  again 
at  the  second  surface,  finally  emerging  approximately  spherical, 
so  that  they  either  converge  to  a  point  I  on  the  side  opposite 
to  0,  as  in  the  figure,  or  diverge  from  a  point  on  the  same  side 
as  0.  Our  task  is  to  derive  a  formula  by  means  of  which, 
knowing  the  distance  of  0  from  the  lens,  the  radii  of  curvature 
of  the  \  two  lens-surfaces,  and  the  index  of  refraction,  we  can 
calculate  the  distance  of  I.  This  might  of  course  be  done  by 
applying  equation  (6)  once  for.  each  surface,  taking  due  ac- 
count of  the  fact  that  the  appropriate  index  to  be  used  at  the 
second  surface  is  the  reciprocal  of  that  for  the  first. 

However,  we  shall  develop  our  lens  formula  by  a  different 
method,  chiefly  because  by  so  doing  we  can  introduce  a  con- 


78  LIGHT 

vention  as  to  algebraic  sign  which  will  prove  more  convenient 
for  our  purpose  than  the  one  used  in  equations  (4)  and  (6). 
We  assume  that  the  lens  is  so  thin  that  its  greatest  thickness 
may  be  neglected  in  comparison  with  the  distance  from  source 
to  image.  Call  the  distance  from  the  lens  to  the  center  of  the 
incident  wavefronts  u,  that  from  the  lens  to  the  center  of  the 
refracted  wavefronts  v.  u  is  considered  positive  when  the 
center  of  the  incident  wavefronts  lies  on  the  side  from  which 
the  light  comes,  that  is,  when  the  incident  light  is  diverging, 
as  is  practically  always  the  case.  Otherwise,  u  is1  negative.  On 
the  other  hand,  we  consider  v  as  positive  when  the  center  of 
the  refracted  wavefronts  is  on  the  side  opposite  to  that  from 
which  the  light  comes,  that  is,  when  the  light  leaves  the  lens 
in  a  converging  beam.  The  radius  of  curvature  of  the  first 
surface  of  the  lens  will  be  considered  positive  when  that  sur- 
face is  convex  to  the  incident  light ;  that  of  the  second  surface 
is  positive  when  it  is  concave  to  the  incident  light.  By  this 
convention,  all  four  of  these  quantities  will  be  positive  m  the 
most  common  case,  viz.,  when  a  double-convex  lens  forms  a  real 
image.  In  figure  42  LAL'B  is  a  somewhat  exaggerated  diagram 
of  a  lens.  0  is  the  center  of.  the  incident  waves,  or  source,  I 
is  the  center  of  the  emergent  waves,  or  image.  Let  rt  be  the 
radius  of  curvature  of  the  first  surface,  LAL',  ra  that  of  the 
second,  LBL'. 


Figure  42 

In  developing  the!  formula,  we  shall  use  the  following 
principle :  In  order  that  I  shall  be  the  image  of  0,  there  must 
be  the  same  number  of  wavelengths  in  every  path  from  0  to  I. 
In  particular,  there  are  as  many  in  the  distance  OL  -f-  LI  as  in 
the  straight  path  OABI.  Of  course  this  can  be  true  only  be- 
cause in  part  of  the  shorter  path,  viz.,  in  the  distance  AB,  the 
wavelength  is  shorter  than  in  the  air.  We  have  already  seen 


LENSES  79 

that,  if  A  is  the  wavelength  in  air,  A/n  will  be  that  in  glass 
whose  index  of  refraction  is  n.  Therefore  the  total  number  of 
wavelengths  in  the  straight  path  is 

(OA  +  BI)/A  +  n  X  AB/A  =  (OA  +  BI  +  n  X   AB)/A 
The  number  in  the  path  OL  +  LI  is 

(OL  +  LI)/A 
Equating  these  two  expressions, 

OL  +  LI  =  OA  +  BI  +  n  X  AB 
or 

OL  —  OA  +  LI  —  BI  =  n  X  AB 

Now  draw  the  arcs  LxL',  from  0  as  center,  and  LyL',  from  I 
as  center.  Then  OL  =  Ox,  and  LI  =  Iy.  Therefore, 

Ox  —  OA  4-  yl  —  BI  =  n  X  AB 
Ax  +  B  y  =  n  X  AB 

But  Ax  is  the  sum  of  the  sagittaVof  the  incident  wavefront 
and  the  first  lens-surface;  By  is  the  sum  of  the  sagittas  of  the 
emergent  wavefront  and  the  second  lens-surface:  and  AB  is 
the  sum  of  the  sagittas  of  the  two  lens-surfaces, — all  with  the 
same  chord  LL'.  Therefore,  we  may  substitute  the  appropriate 
values  obtained  from  equation  (1),  and  get 


2u 
or 


or 


This  is  the  approximate  formula  for  a  lens.  It  is  ad- 
mittedly not  accurate,  and  indeed*  no  perfectly  accurate  formula 
can  be  found.  For  the  wavefront  emerging  from  a  lens  is  not 
accurately  spherical.  Consequently  it  has  no  true  center  and 


1.1          11          /I    ,     1\ 

-  -I 1-    —  4 —n  ( I 

u  •     TL   '     v   '     r2         \rl        T2  I 


80  LIGHT 

there  is  no  perfect  focus.  It  is  possible,  by  giving  the  lens- 
surfaces  a  special  non-spherical  form,  to  make  the  emergent 
wavefronts  really  spherical;  but  this  can  be  done  only  for  a 
certain  fixed  distance  of  0  from  the  lens,  and  the  emerging 
waves  are  no  longer  accurately  spherical  if  the  source  is  moved 
closer  to  the  lens  or  farther  away.  There  is  therefore  no  ad- 
vantage to  be  gained  by  using  the  mathematically  correct 
form,  except  in  the  case  of  telescopic  and  microscopic  objec- 
tives, which  are  always  used  under  the  same  conditions.  With 
spherical  lens-faces,  formula  (7)  is  accurate  enough  for  all 
ordinary  purposes,  provided'  the  lens  is  thin  and  its  diameter  is 
not  more  than  1/20  the  distance  u  or  v.  For  photographic 
lenses  and  microscopic  objectives,  which  are  thick  and  have 
relatively  large  diameters,  it  becomes  very  inaccurate. 

Since  the  right-hand  member  of  (7)  contains  only  terms 
which  are  constant  for  a  given  lens  (rt,  r2,  and  n)  it  is  con- 
venient to  replace  it  by  a  single  symbol,  1/f,  where  f  is  known 
as  the  focal  length  of  the  lens.  The  formula  then  becomes 

-+-=*  (8) 

u        v       f 

which  is  identical  with  one  of  the  forms  of  the  equation  (4). 
The  meaning  of  the  focal  length  f  is  also  the  same  in  the  case 
of  mirror  and  lens.  That  is,  f  is  the  distance  from  the  lens  to 
the  principal  focus,  which  is  the  point  to  which  incident  plane 
waves  would  be  brought  to  focus  by  the  lens,  or  the  point  such 
that  if  the  center  of  the  incident  wavefronts  were  located  there 
the  emergent  wavefronts  would  be  plane.  In  fact,  the  only 
differences  between  (4)  and  (8)  are — first  the  difference  in 
convention  as  to  sign,  already  explained, — second,  the  fact  that 
the  focal  length  of  a  mirror  is  simply  half  the  radius,  while 
that  of  a  lens  is  a  function  of  two  radii  and  an  index  of  re- 
fraction, having  the  value 

f=  (—      *'**     .       .  (9) 

If  we  solve  equation  (8)  for  v,  we  get 

uf 
v  =  1T=f 


LENSES  81 

As  already  noted,  u  is  practically  always  positive,  so  that  if  f 
is  also  positive,  v  will  be  +  if  u>f,  infinite  if  u  =  f ,  and  — - 
if  u  <  f .  That  is,  the  emergent  wavefronts  will  be  convergent 
if  the  source  lies  beyond  the  principal  focus,  plane  if  it  is  at 
the  principal  focus  and  divergent  if  it  lies  between  the  prin- 
cipal focus  and  the  lens  itself. 

A  negative  value  for  f  itself  means  that  plane  waves  fall- 
ing upon  it  from  the  left  would  not  be  converged  to  a  point 
on  the  right,  but  diverged  as  if  they  came  from  a  point  on  the 
same  side  as  the  incident  light.  Equation  (9)  shows  that  in 
order  for  f  to  be  negative  either  rl  and  r,  must  both  be  nega- 
tive, or  the  larger  one  must  be  positive  and  the  smaller  nega- 
tive, since  in  all  practical  cases  n  is  greater  than  1.  This  is 
the  same  as  saying  that  f  is  negative  if  the  lens  is  thinner  in 
the  middle  than  at  the  edges,  positive  if  thicker  at  the  middle 
than  at  thp  edges.  In  the  latter  case  we  say  that  the  lens  is 
converging  or  convex,  in  the  former  case  diverging  or  concave, 
In  figure  43  are  shown  three  different  types  of  converging,  and 
three  of  diverging  lens.  In  order  from  left  to  right,  they  are 
named  planoconvex,  double  convex,  concavoconvex  (or  menis- 
cus), convexoconcave,  double  concave,  and  planoconcave. 


Figure    43 

From  the  elementary  theory  of  lenses  that  we  have  given 
here,  it  is  immaterial  which  face  is  turned  toward  the  incident 
light,  for  equation  (9)  shows  that  rx  and  r2  can  be  interchanged 
without  affecting  the  value  of  f ,  and  such  an  interchange  would 
be  the  only  effect  of  turning  the  lens  around.  That  is,  for 
example,  the  meniscus  type  has  the  same  focal  length  no  mat- 
ter whether  the  convex  or  the  concave  face  be  turned  toward 
the  incident  light.  But  a  more  thorough  study  of  lenses  shows 
that  there  usually  is  a  choice,  depending  upon  the  circumstances 
under  which  the  lens  is  to  be  used.  In  some  cases,  it  is  best 
to  use  a  meniscus  or  planoconvex  lens,  with  the  faces  turned  in 
a  certain  way,  while  in  others  a  symmetrical  double  convex 


82 


LIGHT 


will  function  better,  etc.  The  complete  theory  of  lenses  is  a 
long-  and  difficult  study  in  itself,  and  cannot  be  taken  up  in 
this  book. 


Figure  44 

The  quantity  u  can  never  be  negative  so  long  as  the  lens 
receives  the  light  directly  from  the  source.  But  figure  44  shows 
a  case  where,  for  the  second  lens,  L2,  u  is  negative.  (Here  the 
wavefronts  are  not  drawn,  but  the  course  of  the  light  is  suffi- 
ciently well  indicated  by  the  limiting  rays  of  the  beam.)  0 
is  a  point-source,  the  light  from  which  would  be  brought  by 
lens  Lt  to  a  focus  at  Iit  Lens  L,  therefore  receives  convergent 
light  whose  center  is  at  Ix,  and  in  order  to  find  the  position  of 
the  final  image  L  we  must  substitute  for  u  in  equation  (8)  the 
numerical  value  of  the  distance  L.,!^  with  a  negative  sign  in 
front  of  it,  and  then  solve  as  usual  for  v. 

32.  Two  lenses  in  contact. — We  can  now  prove  that  when 
two  thin  lenses  are  placed  very  close  together  they  act  approxi- 
mately as  a  single  lens,  the  reciprocal  of  whose  focal  length  is 
equal  to  the  sum  of  the  reciprocals  of  the  focal  lengths  of  the 


<  r. 


Figure  45 

two  given  lenses.     See  figure  45.     Let  ft  be  the  focal  length  of 
L,,  f2  that  of  L,.    Aplying  equation  (8)  to  each  lens,  we  get 


- 

u,    •     v,       f, 

i+.L     i 

U2        V2~~  £. 
Adding  these  two  equations,  we  get 


u, 


f, 


CHROMATIC  ABERRATION  83 

Since  the  center  of  the  emergent  wavefronts  for  the  first  lens, 
I,,  is  also  the  center  of  the  incident  wavefronts  for  the  second, 
and  since,  Lx  and  L2  being  very  close  together,  they  are  almost 
the  same  distance  from  I1?  v,  and  u2  are  numerically  practically 
equal,  but  opposite  in  algebraic  sign.  Therefore  l/u2  and  l/vt 
cancel  one  another,  and  we  have  left 

14-   -       1       1 
*i       v2      f,  'hf2 

Uj  for  the  first  lens  is  simply  u  for  the  combination,  and  v2  for 
the  second  is  v  for  the  combination.  Therefore,  if  we  replace 
1/f !  -f-  1/f  2  by  the  single  constant  1/f,  we  get  for  the  combina- 
tion the  ordinary  equation  for  a  single  lens 

1/u  4-  1/v  =  1/f 
where 

K+C  ••  y  <«»  -&; 

The  reciprocal  of  the  focal  length  of  a  lens  is  sometimes  spoken 
of  as  its  dioptric  strength,  and  practical  opticians  adopt  a  lens 
of  one  meter  focal  length  as  the  unit,  calling  it  a  lens  of  one 
diopter.  A  lens  of  two  diopters  would  then  be  one  of  focal 
length  50  cm.,  a  lens  of  %  diopter  one  of  400  cm  focal  length, 
etc.  The  above  demonstration  tlien  shows  that  when  two  thin 
lenses  are  placed  very  close  together  their  dioptric  strengths 
are  added.  This  relation  holds  good  even  if  one  of  the  lenses 
is  diverging,  provided  we  take  the  sum  of  the  reciprocals  of 
the  focal  lengths  in  the  algebraic  sense,  the  focal  length  of  the 
diverging  lens  being  negative.  We  shall  find  the  principle 
very  useful  in  discussing  "achromatic,"  or  color-free,  lenses. 
33.  Chromatic  aberration. — We  have  already  seen  that 
the  index  of  refraction  of  a  substance  is  different  for  different 
wavelengths,  or  colors;  and  since  the  focal  length  depends  upon 
the  index  it  is  obvious  that  a  lens,  unlike  a  mirror,  focusses 
different  colors  at  different  points.  This  is  a  serious  defect  in 
simple  lenses,  and  it  would  be  impossible  to  have  very  effective 
lenses  for  telescopes,  microscopes,  or  cameras,  if  it  were  not. 
possible  to  avoid  it  in  some  degree.  Figure  46  is  a  diagram, 
plotted  to  scale,  which  shows  the  variation  in  focal  length  with 


84  LIGHT 

the   wavelength   of   the   light,    for   two  different  lenses,  one  of 

crown  glass    (dotted  line)    and  one  of  flint    glass  (full  line). 

i  Each  lens  has  a  focal  length 

of  100  inches  for  light  of 
wavelength  .0000589  cm. 
(yellow),  and  the  ordinates 
show  the  differences  between 
this  and  the  focal  length  for 
any  other  wavelength  plot- 
ted as  abscissa.  It  is  seen 
Figure  46  that  the  focal  length  for 

the  blue  differs  by  more  than  an  inch  from  that  for  the  yellow 
in  the  flint  lens,  by  something  less  than  this  in  the  crown. 

This  defect  is  known  as  chromatic  aberration.  The  figure 
shows  that  it  is  greater  for  flint  than  for  crown  lenses.  Not 
only  does  flint  have  a  greater  index  than  crown,  but  its  relative 
dispersion,  that  is,  the  percent  change  in  index  for  a  given 
change  in  wavelength,  is  also  greater.  This  fact  enables  us, 
by  combining  a  crown  converging  with  a  flint  diverging  lens, 
to  produce  a  combination  known  as  an  achromatic  lens,  in 
which,  though  the  focal  length  still  varies  for  different  wave- 
lengths, the  variation  is  relatively  small.  The  plan  adopted  is 
to  figure  the  two  lenses  so  that  the  .focal  length  of  the  com- 
bination is  the  same  for  two  chosen  wavelengths,  say  one  in  the 
brighter  red  and  one  in  the  greenish  blue.  It  will  then  be 
slightly  less  for  wavelengths  intermediate  between  these  two, 
somewhat  greater  for  the  deep  red  and  the  blue  and  violet. 

34.  Achromatic  lenses. — In  order  to  explain  the  production 
of  achromatic  lenses  by  a  concrete  example,  we  shall  calculate 
in  detail  the  radii  of  curvature  for  an  achromatic  of  100  inches 
focal  length.  We  first  choose,  from  a  catalogue  of  optical 
glasses,  two  known  respectively  ias  "S.40,  medium  phosphate 
crown,"  and  "0.335,  dense  silicate  flint."  The  refractive  in- 
dices of  each  of  these  glasses  is  given  for  five  different  locations 
in  the  spectrum,  known  as  the  points  A'  (wavelength  = 
.00007677cm.,  deep  red),  C  (wavelength  =  .00006563cm., 
bright  red),  D  (wavelength  =  .00005893cm.,  orange-yellow),  F 
(wavelength  =  .00004862cm.,  blue-green),  and  G'  (wavelength 
=  .00004341cm.,  deep  blue).  The  table  of  indices  follows: 


ACHROMATIC   LENSES  85 


S.40  (crown) 

0.335  (flint) 

A'  1.55354 

1.62621 

C   1.55678 

1.63197 

D   1.5590 

1.6372 

F   1.56415 

1.65028 

G'  1.56953 

1.66152 

We  are  to  find  what  must  be  the  radii  of  curvature  of  the 
crown  glass  converging  and  the  flint  glass  diverging  lens,  in 
order  that  the  combination  shall  have  a  focal  length  of  100 
inches  for  the  C  and  also  for  the  F  light.  To  simplify  the 
problem,  we  shall  assume  that  the  second  surface  of  the  flint 
lens  is  flat,  and  that  its  first  surface  fits  exactly  over  the 
second  surface  of  the  crown,  so  that  the  combina- 
tion will  appear  like  figure  47,  which  is  a  very 
common  type  of  achromatic.  Then  the  radii  of 
the  four  surfaces,  beginning  with  the  left  hand, 
will  be  indicated  by  a,  b,  — b,  and  o> .  47 

Let  Fc'  and  Fc"  represent  respectively  the  focal  lengths  ot 
the  crown  lens  and  the  flint  lens,  for  the  C  light.  Then,  in 
order  that  the  combination  shall  have  a  focal  length  of  100 
inches  for  C  light,  we  must  have,  by  equation  (10), 

1          1          1 
160~FC'+FC" 

From  equation   (9),  or  from   (7)   and   (8),  we  have  that  for 
any  lens 


therefore,  substituting  the  appropriate  values  of  n,  rl  and  r,, 
for  both  lenses,  we  get 


.jL-  =  _. 63197  jj 

Therefore 

1  /1        1    \  1 

(11) 


86  LIGHT 

Of  course  we  get  an  exactly  analogous  equation  from  the  fact 
that  the  focal  length  of  the  combination  is  also  100  inches  for 
the  F  light,  viz.,  the  equation 

4-56415(1  +  J.)-. 86028  J-  (12) 

In  the  two  equations  (11)  and  (12),  we  may  regard  I/a  +  1/b 
on  the  one  hand,  and  1/b  alone  on  the  other,  as  two  unknown 
quantities,  and  solve  for  their  numerical  values.  The  result  is 

-   4-  -i  =  .037269      4  =  .016998 
an  b 

a  =  49.33  inches.  b=  58.84  inches. 

Therefore,  if  the  crown  lens  be  ground  with  convex  sur- 
faces of  radius  49.33  in.  and  58.84  in.  ! respectively,  and  the 
flint  lens  with  one  surface  concave  of  radius  58.84  in.  and  the 
other  surface  plane,  then  the  combination  will  have  exactly  the 
same  focal  length,  viz.,  100  in.,  for  the  bright  red  and  the  green- 
blue  light.  In  order  to  find  the  focal  length  of  the  combina- 
tion for  other  colors,  we  can  make  the  calculations  very  simply 
by  using  the  already  found  values  for  the  radii,  and  the  appro- 
priate values  for  the'  refractive  indices.  Thus,  if  Fa,  F^,  and 
Fg  represent  the  focal  lengths  for  the  A'  light,  the  D  light,  and 
the  G'  light  respectively, 

*-  =  .55354  (- +  -r  )  —  -62621  ~=  .55354  X  -037269  —  .69621 
X. 016996   Fa  =100.13  in. 

-i-  =  -55900  (-  +  4  W  .63720  4=  -55900  X  .037269  —  .63720 
Fd  \  a  b/  b 

X  .016996  Fd  =  99.94  in. 

^  =  .56953  (-  +  4)  —.66152  -£=  .56953  X  .037269  —  .66152 
.r  g  \  a  b/  D 

X  .016996  Fg  =  100.17  in. 

These  results  are  plotted  in  figure  48,  to  the  same  scale 
used  in  figure  46.  Since  the  lenses  for  which  the  latter  figure 
is  drawn  are  supposed  to  be  ma,de  from  the  identical  glasses 
which  we  have  used  in  our  calculation,  a  comparison  of  the  two 
figures  shows  very  clearly  the  superiority  of  an  achromatic 


IMAGE  OF  AN  EXTENDED  OBJECT  87 

lens  over  a  single-piece  lens  made  from  either  of  the  glasses 

composing  the  achromatic.     Simple  lenses  of  crown  glass  are 

practically    never    used,    ex- 

cept   as    spectacle-lenses,    as 

condensers  for  lantern  or  mi- 

croscopes,   and   in   some   few 

other  cases  where  good  defini- 

tion is  not  required.     Lenses 

are    never    made    from    flint 

glass  alone.    It  would  be  less 

suitable  than  crown,  not  only  Figure  48 

because  of  its  greater  relative  dispersion,  but  also  because  flint 

glasses  are  generally    softer   and    more    easily    scratched    than 

crown. 

Whenever,  as  in  the  example  calculated  above,  the  crown 
and  flint  components  of  an  achromatic  lens  have  one  radius 
of  curvature  in  common,  so  that  they  fit  to  one  another,  they 
are  cemented  together  with  Canada  Balsam.  This  procedure 
prevents  part  of  the  loss  of  light  that  would  otherwise  occur 
by  reflection  at  the  two  surfaces. 

35.  Image  of  extended  object.  Undeviated  ray.  —  Up  to 
this  point  in  our  discussion  of  mirrors  and  lenses,  we  have 
always  supposed  the  source  to  be  located  somewhere  on  the 
axis  of  the  mirror  or  lens.  But  when  we  consider  the  image 
of  an  extended  object  we  must  enquire  what  happens  when  the 
source  lies  off  the  axis,  for  evidently  not  all  points  of  an  object 
of  any  size  can  lie  on  the  axis.  In  this  book  we  shall  not  at- 
tempt to  give  a  mathematical  treatment  of  this  problem,  on 
account  of  its  difficulty,  but  merely  state  the  result  yielded  by 
such  an  investigation,  as  follows:  Let  0  and  I  inl  figure  49 


Figure  49 

be  the  positions  respectively  of  a  source  on  the  axis  of  the  lens 
and  its  image,  as  found  by  formula  (8).  Also  let  0'  be  a  point 
off  the  axis,  but  lying  in  a  plane  perpendicular  to  the  axis 
through  0.  Then,  provided  00'  is  small  compared  to  the  dis- 


88  LIGHT 

tance  from  the  lens,  the  image  of  0',  which  we  shall  call  I',  is 
found  to  lie  very  nearly  in  a  plane  perpendicular  to  the  axis 
through  I.  These  two  planes,  both  perpendicular  to  the  axis, 
and  so  situated  that  a  point  in  one  finds  its  image  in  the  other, 
are  called  confocal  planes.  In  order  to  locate,  in  the  plane 
through  I,  that  particular  point  which  is  the  image  of  0',  we 
reason  as  follows:  Among  all  the  rays  which  diverge  from  0' 
there  will  be  one  which,  striking  the  first  surface  of  the  lens 
near  the  point  where  the  axis  penetrates  it,  will  be  deflected 
into  the  glass  in  such  a  way  that  it  strikes  the  second  surface 
at  the  same  angle  at  which  it  left  the  first.  For  this  ray,  the 
lens  acts  merely  as  a  flat  plate  of  glass  with  parallel  sides,  and 
the  ray  on  emerging  will  take  a  direction  parallel  to  that  which 
it  had  on  entering.  The  desired  point  I'  will  be  the  point 
where  this  ray  strikes  the  confocal  plane  through  I.  The  ray  in 
question  may  be  called  the  undeviated  ray,  for  although  it 
suffers  a  slight  lateral  displacement  in  traversing  the  lens,  its 
direction  is  not  changed.  The  thinner  the  lens,  the  smaller 
will  be  the  lateral  displacement  of  the  undeviated  ray,  and  the 
closer  will  its  point  of  entrance  into  the  lens  and  its  point  of 
exit  coincide  with  the  geometrical  center  of  the  lens.  There- 
fore, for  thin  lenses,  we  find  the  image  of  such  a  point  as  0' 
by  drawing  a  line  from  0'  through  the  center  of  the  lens,  and 
another  through  I  perpendicular  to  the  axis,  their  intersection 
giving  the  location  of  the  image  of  0  to  an  accuracy  sufficient 
for  most  practical  purposes. 

36.  Magnification. — Incidentally,  this  construction  enables 
us  to  find  the  size  of  the  image  of  an  extended  object  such  as 
the  arrow  00'  of  figure  49.  For,  since  OO'  and  II'  subtend 
equal  angles  from  the  center  of  the  lens,  they  must  be  propor- 
tional to  the  distance  from  the  lens.  That  is, 

n/      v  n-n 

00'  "a 

It  is  also  evident  that  if  object  and  image  lie  on  opposite  sides 
of  the  lens,  as  in  figure  49,  the  image  is  inverted,  while  if  they 
lie  on  the  same  side  it  is  erect. 


MAGNIFICATION 


Figure  49  is  drawn  for  a  converging  lens  arranged  to  pro- 
duce a  real  image,  but  the  facts  stated  above  hold  good  whether 
the  lens  be  converging  or  diverging,  the  image  real  or  virtual. 

Similar  conclusions  hold  for  the  reflected  images  from  a 
mirror.  Here  also  we  have  confocal  planes,  such  that  a  point 
in  one  has  its  image  in  the  other,  but  there  is  of  course  no  such 
thing  as  an  undeviated  ray,  since  a  change  of  direction  is 
always  present  in  reflection.  However,  if  we  draw  from  0'  in 
figure  50  the  ray  OT,  to  the  point  where  the  line  01  meets 


Figure  50 

the  mirror,  the  ray  O'P  will  be  reflected  in  the  direction  PI', 
where,  by  the  laws  of  reflection,  the  angles  O'PO  and  I'PI  are 
equal,  and  the  intersection  of  this  line  with  the  plane  confocal 
to  the  plane  of  0'  will  give  the  image  I'.  It  follows  at  once 
that  equation  (13)  holds  for  mirrors  as  well  as  lenses.  But 
in  the  case  of  a  mirror,  the  image  is  inverted  if  it  lies  on  the 
same  side  as  the  object,  erect  if  on  the  opposite  side. 

An  important  application  of  the  principles  just  stated  is 
illustrated  in  figure  51 .'  Suppose  there  are  two  stars,  prac- 
tically at  an  infinite  distance,  in  the  direction  from  the  lens 
indicated  by  the  letters  C  and  D,  the  arrows  indicating  the 


Figure  51 

direction  in  which  the  light  is  propagated  from  them,  in  plane 
waves.  •  If  either  star  lay  on  the  prolongation  of  the  axis  BM, 
its  light  would  be  focussed  at  the  principal  focus  F.  Other- 
wise, if  the  angles  CBM  and  DBM  are  small,  not  more  than 
a  few  degrees,  the  image;  of  each  star  will  lie  in  what  is  called 
the  principal  focal  plane  of  the  lens,  a  plane  through  F  per- 


90  LIGHT 

pendicular  to  the  axis.  For  plane  waves,  every  line  perpen- 
dicular to  the  wavefronts  is  a  ray.  Therefore,  if  we  draw 
through  the  center  of  the)  lens  a  line  perpendicular  to  each  set 
of  incident  wavefronts,  these  will  be  the  undeviated  rays  from 
the  two  stars ;  and  the  points  where  these  lines  meet  the  prin- 
cipal focal  plane,  c  and  d,  will  be  the  images  of  the  two  stars. 
The  angle  cBd,  subtended  by  the  images  from  the  center  of  the 
lens,  is  then  equal  to  the  angle  subtended  by  the  stars  them- 
selves from  the  center  of  the  lens — or  indeed  from  any  terres- 
trial point  since  the  distance  of  the  stars  is  so  great.  This 
angle  is  given  in  radian  measure,  to  a  sufficiently  close  approxi- 
mation, as  the  quotient  of  the  distance  cd  divided  by  the  focal 
length. 

This  principle  is  used  in  practical  astronomy  for  measur- 
ing the  angular  distance  between  double  stars.  There  are  two 
methods  of  measuring  the  distance  cd  between  the  images. 
One  is  to  place  a  photographic  plate  directly  in  the  focal  plane, 
expose  it  to  the  light  from  the  stars,  and  then  develop  it  by 
the  usual  photographic  processes,  which  leave  a  little  blackened 
dot  where  each  star-image  falls.  The  distance  between  these 
dots  is  accurately  measured  on  a  dividing-engine. 

37.  Micrometer. — The  other  method  is  to  use  a  micrometer, 
the  essential  part  of  which  is  a  small  metal  frame  arranged 
so  that  it  can  slide  in  a  plane  perpendicular  to  the  axis'  of 
the  lens.  A  fine  spider-thread  is  stretched  across  this  frame, 
so  that  it  lies  in  the  focal  plane,  perpendicular  to  the  direction 
in  which  the  frame  slides.  The  whole  micrometer  is  turned 
about  the  axis  of  the  lens,  till  the  direction  in  which  the  frame 
slides  is  parallel  to  the  line  joining  the  two  star-images,  and 
the  frame  is  then  moved  by  a  fine-pitched  screw  to  which  it 
is  attached,  so  that  the  spider- thread,  commonly  called  the 
cross-hair,  lies  first  on  one  image,  then  on  the  other.  The  pitch 
of  the  screw  is  known,  so  that  the  number  of  its  revolutions 
necessary  to  move  the  cross-hair  from  one  image  to  the  other 
gives  the  distance.  In  order  to  make  the  cross-hair  and  star- 
images  clearly  visible,  a  short-focus  lens,  or  combination  of 
Itmses,  called  the  eyepiece,  is  placed  just  behind  the  focal  plane. 
The  eye  sees  magnified  images  of  the  cross-hair  and  the  two 
original  star-images. 


DEFECTS  OF  MIRRORS  AND  LENSES  91 

A  similar  method  is  used  in  certain  surveying  instruments, 
although  the  conditions  are  somewhat  different.  The  objects 
observed  are  not  infinitely  distant,  and  the  images  are  con- 
sequently not  formed  exactly  in  the  principal  focal  plane. 

38.  Imperfections  of  mirrors  and    lenses. — The    student 
will  no  doubt  have  drawn  for  himself  the  conclusion  that,  quite 
apart  from    chromatic    aberration    in  lenses,    both    lenses    and 
mirrors  are  far  from  perfect  optical    instruments,    since    our 
formula?  are  only  approximations.     Such  a  conclusion  is  un- 
doubtedly correct,  and  it  will  be  worth  while  to  enumerate  the 
more   common  faults.  .  We   shall   speak  principally   of  lenses, 
but  what   is   said   applies-  also   to   mirrors,   for  all  the   faults 
found  in  lenses,  except  those  due  to  chromatic  aberration  and 
absorption,  are  also  shared  by  mirrors,  in    many    cases    to    a 
greater  degree. 

In  the  first  place,  owing  to  the  fact  that  light  is  a  wave 
motion,  and  therefore  does  not  travel  absolutely  in  straight 
lines,  no  optical  instrument,  whether  it  be  made  up  of  lenses, 
mirrors,  or  other  elements,  and  no  matter  how  perfect  the 
workmanship  may  be.  can  produce  from  a  point  source  an 
image  which  is  a  real  mathematical  point.  For  instance,  al- 
though a  star,  on  account  of  its  great  distance,  may  be  regarded 
as  a  point  source,  its  image  as  produced  by  the  most  perfect 
telescope  is  not  a  mathematical  point,  but  a  very  small  disc 
surrounded  by  a  series  of  faint  rings.  If  the  lens  is  well  made 
and  of  large  diameter,  the  diameter  of  the  disc  and  the  sur- 
rounding rings  is  so  small  that  the  latter  can  be  seen  only  by 
highly  magnifying  them,  and  they  become  a  source  of  trouble 
only  in  the  most  exacting  work  with  telescope  or  microscope. 
The  nature  of  this  fault  will  be  considered  later  under  the 
head  of  diffraction,  sections  72  and  73. 

39.  Spherical  aberration. — Another  fault    is    known    as 
spherical    aberration.      Quite    apart    from    the    just-mentioned 
difficulty  of  diffraction,   and  from    chromatic    aberration,    the 
rays  coming  through  the  edges  of  a  lens  are  not  brought  to  the 
same    focus   as    those    coming   through   near  the  center.     This 
follows  from  the  fact  mentioned  above,  that  when  a  spherical 
wavefront  is  refracted  at  a  spherical  surface,  it  emerges  not 
truly  spherical.     Figure  52  illustrates  this  defect  in  an  exag- 
gerated manner.     The  rays  are  drawn,  but  not  the  wavefronts. 


92 


LIGHT 


0  is  a  point  source,  from  which  all  the  rays  originate.  When 
they  emerge  from  the  lens,  they  do  not  converge  to  a  single 
point.  Since  the  central  part  of  the  emergent  wavefront,  say 
the  part  to  which  rays  between  those  marked  5  and  7  belong, 


Figure   52 

is  very  nearly  spherical,  these  rays  will  all  intersect  nearly  at 
a  single  point  f.  Rays  4  and  5,  however,  cross  before  reaching 
f;  at  such  a  point  as  e,  and  the  corresponding  rays  7  and  8  at 
e'.  Rays  3  and  4  will  cross  still  nearer  the  lens,  as  at  c,  3 
and  2  at  b,  2  and  1  at  a,  etc.  Consequently,  instead  of  having 
a  single  point  f  as  the  image  of  0,  we  may  say  that  the  image 
is  the  line  abcefe'c'b'a',  or  rather,  the  surface  formed  by  re- 
volving this  line  about  the  axis  of  the  lens.  This  surface  is 
roughly  conical,  with  a  point  or  "cusp"  at  f,  and  by  far  the 
greater  part  of  the  light  is  concentrated  at  this  point,  which 
we  commonly  regard  as  the  proper  image.  Nevertheless,  much 
light  fails  to  pass  through  f,  and  if  a  screen  were  placed  at 
that  point  we  should  see  a  sort  of  halo  surrounding  the  bright 
center,  caused  by  light  which  came  to  focus  before  reaching 
the  screen.  The  curve  abcefe'c'b'a'  is  called  a  caustic,  and  the 
corresponding  surface  a  caustic  surface.  A  familiar  example  of 
a  caustic  is  the  socalled  "cow's  hoof"  seen  on  the  ^urf ace  of 
a  glass  of  milk.  It  is  formed  by  reflection  from  the  inner  sur- 
face/ of  the  rim  of  the  glass,  which  acts  as  a  concave  cylindrical 
mirror,  reflecting  light  from  a  nearby  window,  or  any  other 
conveniently  placed  source  of  light. 

40.  Curvature  of  field. — Still  another  defect  is  curvature 
of  the  field.  Referring  to  figure  49,  it  was  stated  in  the  text 
that  if  0'  lies  in  the  plane  of  0,  its  image  I'  is  very  nearly  in 
the  plane  of  I,  provided  that  the  distance  00'  is  small  com- 
pared to  the  distances  of  0  and  0'  from  the  lens;  and  the 
two  planes  perpendicular  to  the  axis  were  called  confocal 
plane.  More  accurately,  the  surface  which  is  confocal  to  the 


ASTIGMATISM 


93 


plane  through  O  is  not  a  plane  but  a  slightly  curved  surface, 
concave  toward  the  lens  and  nearly  plane  in  the  neighborhood 
of  the  axis.  As  a  special  case,  suppose  O  '  and  0'  are  so  far 
away  that  the  waves  reaching  the  lens  from  them  are  prac- 
tically plane.  We  may  then  regard  0,  0',  and  all  other  suf- 
ficiently distant  objects  as  being  in  a  plane  perpendicular  to 
the  axis  of  the  lens.  Under  these  circumstances,  if  a  screen 
be  placed  perpendicular  to  the  axis  at  the  principal  focus, 
those  of  the  distant  objects  which  subtend  only  a  small  angle 
with  the  axis  will  be  sharply  in  focus  on  the  screen,  but  the 
others  will  be  blurred,  and  the  screen  must  be  moved  closer  to 
the  lens  to  bring  them  into  sharp  focus. 

41.  Astigmatism. — This  is  a  fault  which  shows  itself  par- 
ticularly for  pencils  of  light  which  strike  a  lens  or  mirror 
diagonally.  Under  such  circumstances,  the  image  of  a  true 
point  tends  to  become  a  pair  of  short  lines,  perpendicular  to 
one  another,  but  not  intersecting.  Figure  53  is  intended  to 


Figure  53 

make  this  plain.  The  ellipse  L  represents  a  lens  seen  in  per- 
spective. For  convenience  in  explanation,  suppose  an  opaque 
piece  of  paper  pasted  over  the^  face  of  the  lens,  with  a  square 
hole,  so  that  the  beam  that  comes  through  is  limited  to  what 
passes  through  this  square  aperture.  Only  the  rays  coming 
from  the  four  corners  are  shown.  Rays  A  and  B  intersect  at 
the  point  R,  but  A  and  C  intersect  at  P,  nearer  the  lens. 
Similarly,  C  and  D  intersect  at  S,  B  and  D  at  Q.  Rays  B 
and  C  do  not  intersect  at  all,  neither  do  A  and  D.  There  will 
be  a  horizontal  focal  line  PQ,  for  the  intersection '  of  rays  in 
a  vertical  plane,  and  a  vertical  focal  line  SR,  for  the  inter- 


94  LIGHT      . 

section  of  rays  in  a  horizontal  plane.  The  two  focal  lines  can 
be  shown  very  nicely  by  holding  a  converging  lens,  or  better 
still  a  concave  mirror,  so  that  it  receives  somewhat  obliquely 
the  light  from  the  sun,  and  moving  a  white  card  back  and  forth 
till  the  two  focal  lines  are  found. 

42.  Lenses  for  special  purposes. — In  spite  of  this  long  list 
of  faults,  lenses  function  in  a  very  satisfactory  manner  for 
most  purposes.  Consider  for  instance  the  lenses  used  as 
objectives  in  telescopes.  (Telescopes  are  considered  in  detail 
in  the  following  chapter.)  Chromatic  aberration  is  hardly 
perceptible  if  the  lens  is  properly  constructed  of  flint  and 
crown  glass  in  the  manner  already  described,  for  the  focal 
length  is  very  nearly  the  same  for  all  wavelengths  except  the 
blue  and  violet;  and  since  these  colors  in  most  light-sources 
have  feeble  luminosity,  their  being  somewhat  out  of  focus 
hardly  affects  the  sharpness  of  the  images.  Spherical  aber- 
ration is  negligible  because  the  area  of  the  lens  is  made  small 
enough  so  that  only  that  part  of  the  emergent  wavefront  is 
used  which  is  nearly  spherical.  As  a  rule,  the  diameter  of  a 
telescope  objective  is  somewhere  between  1/30  and  1/16  of  the 
focal  length,  making  the  angular  diameter  of  the  cone  of  light 
which  it  transmits  relatively  small.  Finally,  the  curvature  of 
the  field  and  astigmatism  produce  a  negligible  effect,  on  ac- 
count of  the  smallness  of  the  field.  In  astronomical  telescopes, 
it  is  seldom  necessary  to  use  a  field  of  as  much  as  one  degree. 
Consequently,  only  the  flatter  portion  of  the  field  is  used,  and 
no  pencil  of  light  that  is  visible  in  the  eyepiece  traverses  the 
lens  with  enough  obliquity  to  cause  appreciable  astigmatism. 
-The  design  of  a  telescope  objective  is  therefore  relatively  sim- 
ple, and  its  excellence  depends  mainly  on  the  quality  of  the 
workmanship  and  the  homogeneity  of  the  glass.  This  latter 
condition  is  by  no  means  easy  to  fulfill  in  such  large  discs  of 
glass  as  were  used  in  making  the  objective  of  the  Lick  tele- 
scope (36  inches  in  diameter)  or  that  of  the  Yerkes  (40  inches). 

Camera  lenses  are  used  under  more  exacting  circumstances. 
In  order  that  the  lens  may  be  "fast,"  that  is,  give  sufficient 
illumination  with  very  short  exposure-time,  it  must  have  a 
diameter  as  great  as  1/5  or  1/6  the  focal  length,  giving  great 
opportunity  for  spherical  aberration.  Moreover,  the  extent  of 
field  used  is  large,  since  the  dimensions  of  the  photographic 


LENSES  FOR  SPECIAL  PURPOSES  95 


plate  are  nearly  as  great  as  the  focal  length.  Consequently, 
troubles  due  to  curvature  of  the  field  and  astigmatism  are 
likely  to  appear.  The  manufacturers  of  photographic  lenses 
have,  however,  achieved  remarkable  success  in  combating  these 
difficulties,  so  that  a  first-class  lens  shows  them  to  only  a  limited 
extent.  Even  with  the  best  lenses,  however,  if  they  are  used 
with  full  aperture,  the  corners  of  the  picture  are  slightly  out 
of  focus  due  to  curvature  of  field,  and  show  a  slight  drawing 
out  of  points  into  lines,  which  is  the  result  of  astigmatism.  A 
good  photographic  lens  is  made  in  two  parts,  separated  by  an 
air-space,  and  each  part  is  composed  of  several  pieces  of  glass. 
It  is  by  altering  the  composition  of  these  separate  pieces  and 
the  curvature  of  their  surfaces  that  the  designers  have  suc- 
ceeded in  reducing  largely,  but  not  entirely,  the  inherent  lens- 
defects.  A  photographic  lens  cannot  properly  be  regarded  as 
a  thin  lens. 

Microscopic  objectives  also,  if  of  high  power,  are  objects 
of  elaborate  design,  consisting  of  many  pieces  of  glass.  The 
thickness  of  such  a  so-called  lens  (really  it  is  a  combination  of 
a  number  of  lenses)  is  much  greater  than  the  equivalent  focal 
length  of  the  combination. 

Problems. 

1.  If  a  plate- glass   window,   index   1.58,   appears  to   one 
looking  into  it  to  be  8  mm.  thick,  what  is  the  actual  thickness? 

2.  What  must  be  the  radius  of  curvature  of  a  symmetrical 
converging  lens  of  crown  glass,  to  have  a  focal  length  1  meter  f 

3.  An  object  is  4  ft.  from  a  white  screen.    Find  two  posi- 
tions in  which  a  lens  of  8  inch  focus  can  be  placed,  to  form  an 
image  of  the  object  on  the  screen. 

4.  An  object  is  8  inches  from  a  screen.     Where  should  a 
concave  mirror  of  2  foot  radius  be  placed  to  form  an  image  of 
the  object  on  the  screen. 

5.  Show  that  problem  3  cannot  be  solved  if  the  focal  length 
of  the  lens  is  more  than  12  inches. 

6.  A  lens  of  3  foot  focus  forms  images  of  two  stars  in  its 
principal   focal   plane,   and   a  micrometer   is  used   to   find  tho 
distance  between  the  images.     It  takes  12.85  turns  of  the  screw 
to  move  the  cross-hair  from  one  image  to  the  other,  and  the 


96  LIGHT 

screw  has  50  threads  to  the  inch.    Find  the  angle  between  the 
stars. 

7.  Show  how  an  achromatic  diverging  lens  can  be  made, 
aad  write  the  equations  from  which  the  curvature  of  its  sur- 
faces can  be  found,  using  the  data  for  the  glasses  given  in 
paragraph  38. 

8.  In  what  position  must  the  eye  be    placed,    to    see    an 
image  formed  by  a  lens  ox*  mirror,  if  a  screen  is  not  used? 
Why  is  it  usually  easier  to  find  a  virtual  than  a  real  image? 

9.  If  a  camera  lens  has  a  focal  length  of  8  inches,  find 
the  proper  position  for  focus  on  an  object  5  ft.  away,  and  the 
length  of  this  image  if  the  object  is  2  ft.  long. 

10.  Show  that  if  a  camera  of  focus  6  inches  is  focussed 
for  infinitely  distant  objects,  any  object  more  than  40  ft.  away 
will  be  less  than  5/64  inch  out  of  focus. 

11.  Explain  why  " depth  of  focus"  in  a  camera  is  impos- 
sible to  obtain  without  sacrificing  "speed." 


CHAPTER  VI. 

43.  The  telescope. — 44.  Magnifying  power. — 45.  Ramsden  eyepiece. 
— 46.  Opera  glass. — 47.  Prism  binocular. — 48.  Reflecting  telescopes. — 
49.  Simple  microscope. — 50.  Compound  microscope. — 51.  Projection  lan- 
terns. 

43.  The  telescope. — The  essential  part  of  a  telescope  is 
two  lenses, — a  long-focus,  large  diameter  achromatic,  turned 
toward  the  object  in  view  and  therefore  known  as  the  objective, 
and  a  smaller  lens  (or,  as  we  shall  see  later,  more  commonly 
a  pair  of  lenses)  called  the  eyepiece.  Figure  54  shows  a  sim- 
ple diagram  of  a  telescope.  The  object  viewed  is  supposed  to  be 
an  arrow,  very  far  away,  but  so  large  that  in  spite  of  distance 
it  covers  an  angle  of  a  degree  or  so.  If  this  conception  seems 
too  artificial,  we  may  think  of  the  point  of  the  arrow  as  repre- 
senting one  star,  the  butt  another.  Wavefronts  are  not  indi- 
cated, but  lines  are  drawn  to  show  the  course,  through  the  in- 
strument, of  the  cone  of  light  from  each  end  of  the  arrow. 
Dotted  lines  show  the  undeviated  rays  for  each  lens. 


A  real  inverted  image  of  the  object  is  formed  in  the  focal 
plane  of  the  objective,  from  which  the  waves  continue  on, 
diverging  from  this  image  exactly  as  if  it  were  a  material 
object,  except  that  the  light  is  limited  to  a  comparatively  small 
cone.  This  light  falls  upon  the  eyepiece,  which  forms  with  it 
a  second  image,  really  an  image  of  an  image.  Since  the  rays 
that  form  any  point  of  the  first  image  are  limited  to  the  cone 
that  comes  through  the  objective,  it  may  well  happen  that  the 
undeviated  ray  drawn  from  this  point  through  the  center  of 
the  eyepiece  lies  outside  the  cone  and  therefore  does  not  exist 
as  a  real  ray.  But  the  position  of  the  second  image  must  cer- 

(97) 


98  LIGHT 

tainly  be  independent  of  the  diameter  of  the  objective,  and 
therefore  we  are  at  liberty  in  such  a  case  to  find  that  position 
by  drawing  fictitious  undeviated  rays  just  as  if  they  really 
did  exist.  The  figure  is  drawn  for  such  a  case.  The  position 
of  the  second  image  depends  of  course  upon  the  location  of 
the  eyepiece  which  is  mounted  so  that  the  observer  can  slide 
it  at  will  through  a  short  distance  toward  or  away  from  the 
objective.  Most  observers  place  it  so  that  the  first  image  lies 
a  little  within  its  principal  focus.  Then  the  second  image,  the 
one  which  the  eye  sees,  is  virtual,  still  inverted,  and  on  the 
same  side  as  the  first  image,  but  farther  away.  This  is  shown 
by  AB  in  the  figure.  If  the  principal  focus  x  were  placed  just 
at  the  real  image  ab,  as  is  sometimes  done  by  persons  of  far- 
sighted  or  normal  vision,  AB  would  be  thrown  back  to  infinity, 
like  the  original  object,  but  would  still  subtend  a  much  greater 
angle  than  the  latter. 

44.  Magnifying  power. — We  take  as  a  measure  of  the  mag- 
nifying power  of  the  telescope  the  ratio  of  the  angle  subtended 
by  the  image  AB  to  that  subtended  by  the  original  object,  and 
in  calculating  its  numerical  value  we  assume,  for  the  sake  of 
definiteness,  that  the  principal  focus  of  the  eyepiece  coincides 
exactly  with  that  of  the  objective,  that  is,  that  x  lies  exactly 
on  ab,  putting  AB  at  an  infinite  distance.  With  both  the 
original  object  and  the  final  image  so  far  away,  it  does  not 
matter  what  point  is  chosen  as  the  apex  of  the  angles  sub- 
tended. That  subtended  by  the  object  is  aOb,  which,  in  radian 
units,  has  the  value  ab/F,  F  being  the  focal  length  of  the 
objective.  That  subtended  by  the  image  AB  is  ApB,  whose 
value  is  ab/f,  f  being  the  focal  length  of  the  eyepiece.  There- 
fore the  magnifying  power  is 

ab 

f        F1 
z  —  L 

—  — i IT 

ab       f 
F 

Therefore,  for  high  magnifying  power,  we  should  use  a  long- 
focus  objective  and  a  short-focus  eyepiece.  Usually,  a  large 
telescope  is  provided  with  several  eyepieces  of  different  focal 
length,  so  that  the  magnifying  power  can  be  changed  at  will. 
For  some  purposes,  high  magnification  is  desirable,  for  others 


MAGNIFYING  POWER  OF  TELESCOPES  99 

not.  The  higher  the  magnification,  the  smaller  the  visible  field: 
that  is  the  smaller  the  area  that  can  be  seen  at  once.  For 
instance,  with  high  magnification  only  a  very  small  part  of  the 
surface  of  the  sun  or  moon  can  be  seen. 

There  are  other  practical  limitations  to  the  magnifying 
power  that  can  be  used  with  advantage.  As  we  have  previous- 
ly stated,  the  real  imag'e  produced  by  the  objective  is  not  a 
true  picture  of  the  object,  but  is  slightly  hazy  at  the  edges,  and 
is  surrounded  by  faint  diffraction  rings.  Any  increase  in  mag- 
nification beyond  the  point  where  these  rings  or  bands  become 
visible  is  useless,  for  it  merely  magnifies  the  bands  and  the 
haziness  along  with  the  rest  of  the  image  and  contributes  noth- 
ing to  distinctness  of  vision.  Astronomers  also  find  that  cer- 
tain conditions  of  the  atmosphere,  caused  no  doubt  by  irregu- 
larities in  density,  produce  a  haziness  or  fuzziness  of  image 
known  as  "bad  seeing,"  which  is  worse  than  the  diffraction 
difficulty.  In  fact,  with  a  large  telescope,  the  appearance  of 
the  diffraction  bands  oh  high  magnification  indicates  that  the 
"seeing  is  good,'7  for  poor  seeing  conditions  cause  them  to  be 
blurred  out  of  recognition.  With  only  moderately  good  seeing, 
an  astronomer  will  use  moderate  magnification,  and  with  very 
bad  seeing  he  will  abstain  from  observing  at  all. 

Figure  54  shows  that  at  a  certain  place  cd  the  two  cones 
of  light  from  the  head  and  the  butt  of  the  arrow  cross.  In 
fact,  there  is  a  little  circle  at  this  place  through  which  passes 
every  cone  of  light  that  traverses  the  telescope,  and  it  Is  not 
hard  to  show  that  this  circle  is  nothing  more  nor  less  than  the 
image  of  the  objective  lens  formed  by  the  eyepiece.  For  best 
vision,  the  eye  should  be  held  so  that  its  pupil  coincides  with 
this  small  circle,  which,  by  the  way,  is  called  the  exit -pupil  of 
the  telescope.  For  the  figure  shows  clearly  that  if  the  eye  is 
held  much  closer  to  the  eyepiece,  or  much  farther  from  it,  only 
cones  of  light  from  the  middle  part  of  the  arrow  (cones  not 
drawn  in  the  figure)  would  enter  the  pupil  of  the  eye,  unless 
the  latter  were  very  large.  To  see  the  whole  image  AB  at  once 
would  then  be  impossible,  though  one  could  see  different  parts 
of  it  at  a  time  by  moving  the  eye  up  or  down,  so  as  to  receive 
the  light  from  those  parts.  That  part  of  the  image  that  can 
be  seen  in  any  one  position  of  the  eye  is  called  the  field  of  view, 


100  LIGHT 

and  it  is  greatest  when  the  pupil  of  the  eye  coincides  with  the 
exit-pupil.  If  the  eye  is  in  thisi  most  favorable  position,  the 
field  of  view  is  still  limited  by  the  diameter  and  the  position  of 
the  eyepiece.  Reference  to  the  figure  shows  that  if  the  arrow 
were  much  larger  than  it  is  there  made,  the  cones  of  light  from 
the  ends  of  the  image  would  partly  or  wholly  miss  the  eye- 
piece. If  they  missed  the  latter  entirely,  these  ends  would  be 
completely  invisible  in  the  telescope,  while  if  only  part  of  the 
cone  fell  on  it,  the  image  would  be  faint  at  the  ends.  The 
simple  eyepiece  shown  in  the  figure  is  not  suitable  for  produc- 
ing a  large  and  uniformly  illuminated  field  of  view.  Jt  is 
desirable  that  the  eyepiece  should  come  very  close  to  the  real 
image  ab;  but  in  order  to  have  this  occur  with  a  single-lens 
eyepiece,  the  focal  length  of  the  latter  would  have  to  be  inordi- 
nately short;  and  if,  in  addition,  the  diameter  were  made 
great,  spherical  aberration  and  other  lens  defects  would  become 
too  pronounced. 

45.  Rainsden  eyepiece. — For  the  reasons  outlined  above, 
much  ingenuity  has  been  applied  in  devising  eyepieces  which 
consist  of  combinations  of  lenses  instead  of  single  lenses,  so  as 
to  give  a  larger  field  for  the  same  magnifying  power.  The  best 
known  of  these  is  the  Ramsden  eyepiece,  which  functions  very 
satisfactorily,  and  is  used  on  most  telescopes.  It  is  shown  in 
figure  55.  The  objective  of  the  telescope  has  been  omitted, 

from  the  drawing,  but  the  real 
image  ab  and  cones  of  light 
forming  its  ends  are  shown  just 
as  they  are  in  figure  54.  The 
eyepiece  consists  of  two  identi- 
cal planoconvex  lenses,  mounted 
rigidly  in  a  metal  tube,  and  separated  by  a  distance  equal  to 
%  the  focal  length  of  either.  It  can  be  shown  that  such  a  pair- 
is  equivalent,  so  far  as  magnification  is  concerned,  to  a  single 
lens  whose  focal  length  is  %  that  of  either  component.  The 
front  lens,  called  the  field-lens,  is  placed  very  close  to  the  real 
image  ab,  and  therefore  forms  from  it  a  virtual  image  a'b', 
slightly  larger,  and  slightly  farther  away.  In  fact,  a'b',  comes 
just  at  the  principal  focus  of  the  rear  lens  of  the  combination 
(called  the  eye-lens)  or  just  within  it.  Accordingly,  this  latter 
forms  the  final  virtual  image,  AB  of  figure  54,  either  at  infinity 


OPERA  GLASS  101 

or  at  whatever  distance  from  the  eye  is  most  suitable  for  the 
observer,  adjustment  being  secured  by  sliding  the  whole  eye- 
piece toward  or  away  from  the  real  image  ab.  Since  ab  lies 
close  to  the  field  lens,  the  latter  receives  and  transmits  light 
from  an  area  of  the  image  practically  equal  to  the  area  of  the 
lens  itself,  thus  giving  a  field  whose  diameter  is  approximately 
equal  to  the  diameter  of  the  field-lens.  The  pencils  from  all 
parts  of  the  image  cross  the  axis  just  behind  the  eye-lens.  This 
is  the  most  convenient  place  for  the  exit-pupil,  for  the  eye  can 
then  be  placed  close  up  to  the  end  of  the  eyepiece.  It  is  found 
further  that  this  combination  of  two  planoconvex  lenses  ia 
almost  free  from  spherical  aberration,  and  the  chromatic  aber- 
ration is  of  such  a  nature  asi  to  be  hardly  perceptible. 

If  a  micrometer  is  used  with  the  telescope,  it  is  placed  so 
that  the  crosshairs  move  exactly  in  the  plane  ab  in  which  the 
real  image  lies.  The  proper  method  of  adjusting  the  telescope, 
known  as  focussing,  is  as  follows :  First,  the  eyepiece  is  pushed 
in  or  out  until  the  crosshairs  are  visible,  clearly  and  without 
eyestrain.  Then,  by  means  of  a  suitable  slide  in  the  telescope 
tube,  the  eyepiece  and  crosshairs  together  are  pushed  in  or  out 
till  the  image  of  the  object  viewed  is  also  seen  clearly,  and  there 
is  no  parallax  between  the  image  and  the  crosshairs. 

One  defect  of  the  kind  of  telescope  we  have  been  describ- 
ing is  that  the  image  is  inverted.  This  is  not  a  disadvantage 
in  astronomical  telescopes,  but  for  terrestrial  telescopes  it  is 
inconvenient.  In  such  instruments  the  image  is  usually  rein- 
verted  by  making  the  tube  of  the  instrument  very  long,  and 
inserting  between  the  eyepiece  and  the  real  image  formed  by 
the  objective  another  lens  or  pair  of  lenses  of  rather  short 
focal  length,  whose  function  is  to  receive  the  light  from  the  real 
image  (ab  of  figure  54)  and  form  therefrom  another  real  image 
which  is  reinverted  and  therefore  right  side  up.  The  eye- 
piece then  forms  from  this  image  a  magnified  virtual  image 
which  the  eye  sees.  The  great  length  of  tube  necessary  in  this 
form  of  instrument  makes  it  inconvenient  except  for  small  spy- 
glasses. 

46.  Opera  glass. — In  the  old-fashioned  opera  glass,  shown 
in  diagram  in  figure  56,  the  erection  of  the  image  is  provided 
for  by  using  a  diverging  lens  for  the  eyepiece.  This  form  of 
telescope  is  quite  short;  for,  in  order  that  the  eyepiece  may 


102 


LIGHT 


magnify  the  image,  it  is  necessary  that  it  should  intercept  the 
light  between  the  objective  and  the  latter 's  principal  focus. 
Thus  the  real  image1  ab  is  not  actually  formed  at  all,  for  it 
would  come  behind  the  eyepiece  instead  of  before  it.  Con- 


Figure  56 

verging  waves  strike  the  eyepiece,   with  their  centers  on   ab. 
Therefore,  in  the  formula 


or  its  equivalent 


uf 


-u  — f 


a  must  be  taken  as  negative.  Since  f  is  also  negative,  because 
the  lens  is  diverging,  the  numerator  of  the  fraction  in  the  last 
equation  is  positive,  and  the  sign  of  v  depends  upon  the  sign 
of  u  —  f ,  where  both  u  and  f  are  essentially  negative.  In  order 
to  have  a  virtual  image  AB,  v  must  be  negative,  therefore  u 
must  be  greater  in  absolute  amount  than  f ;  that  is,  the  prin- 
cipal focus  of  the  eyepiece  must  lie  between  the  latter  and  the 
place  where  the  image  ab  would  come  if  the  lens  were  removed 
For  example,  let  f  =  -  —  5cm.,  u  =  -  —  5.2cm.  Then  v  = 
— 130  cm.  The  observed  image  AB  would  then  lie  130  cm. 
away  on  the  side  from  which  the  light  comes,  and  would  be 
virtual.  Also,  it  would  be  inverted  as  regards  ab,  erect  as 
regards  the  original  object.  The  erectness  of  image,  and  its 
shortness  make  this  type  of  telescope  convenient,  but  unfor- 
tunately its  field  is  quite  small.  The  diagram  shows  that  there 
is  no  real  exit-pupil,  as  there  is  in  the  ordinary  form  of  tele- 
scope; that  is  there  is  no  place  where  the  eye  can  be  placed 
so  that  it  will  receive  every  cone  of  light  that  does  not  miss 


PRISM  BINOCULAR.  REFLECTING  TELESCOPE  103 

the  eyepiece.  The  eye  must  be  moved  about  in  order  to  see 
the  whole  image  of  any  object  viewed,;  unless  the  object  be 
relatively  small.  For  this  reason,  such  instruments  are  made 
only  with  small  magnifying  powers,  say  two  or  three  diameters. 

47.  Prism  binocular. — The  modern  prism-binocular  is  a 
great  improvement  over  the  form  of  telescope  described  above. 
It  is  essentially  a  telescope  of  the  form  described   (or  rather, 
a  pair  of  such  telescopes,   one   for  each   eye),   in   which   the 
length  is  very  much  reduced  by  four  reflections  in  totally  re- 
flecting prisms.     Incidentally,  the  series    of    reflections    rein- 
verts  the  image,  so  that  it  is  possible  to  use  an  eyepiece  like 
the  Ramsden,  with,  its  large  field.     An  additional  advantage 
lies  in  the  fact  that  the  arrangement    of    the    two    telescopes 
brings  the  objectives  farther  apart  than  the  two  eyes.     This, 
being  equivalent  to  a  wider  spacing  of  the  eyes  themselves, 
greatly  increases  the  stereoscopic  effect,  or  parallax,  and  brings 
the  field  into  strong  and  pleasing  relief. 

48.  Reflecting  telescopes. — A  telescope  composed  of  ob- 
jective lens  and  eyepiece  is  known,  among  astronomers  as  a 
refractor.    A  reflector  is  a  telescope  in  which  a  concave  mirror 
is  substituted  for  the  objective  lens.     At  the  present  time  the 
use  of  reflectors  is  confined  almost  exclusively  to  photographic 
work,  for  which  purpose  they  possess  several  decided  advan- 
tages.    In  the  first  place,  they  are  free  from  chromatic  aberra- 
tion.   Secondly,  they  have  no  absorption,  and  this  is  very  im- 
portant in  photography,  for  the  transmission  of  light  through 
glass    causes   much    of   the    photographically  active  ultraviolet 
light  to  be  lost}  by  absorption.     Finally,  by  making  the  con- 
cave reflecting  surface  parabolic  instead  of  spherical,  the  prin- 
cipal focus  is  rendered  absolutely  free  from  spherical  aberra- 
tion, and  a  small  region  in  its  neighborhood  almost  so,  so  that 
most  beautiful  definition  is  secured    in    photographing   objects 
of  such  small  angular  dimensions  as  a  star-cluster  or  a  small 
nebula. 

The  concave  mirror  is  usually  placed  at  one  end  of  a  long 
tube  or  frame  work,  the  other  end  of  which  is  open  and  pointed 
toward  the  celestial  body  to  be  photographed.  Between  the 
mirror  and  its  principal  focus,  is  placed  a  small  plane  mirror, 
set  at  an  angle  of  45°  with  the  axis  of  the  instrument.  This 
reflects  the  light  coming  from  the  concave  mirror  to  the  side 


104 


LIGHT 


of  the  tube,  where  the  photographic  plate  is  exposed  in  the 
reflected  position  of  the  focal  plane.  A  Ramsden  eyepiece  may 
also  be  placed  in  position  to  receive  the  light,  but  this  has  no 
function  during  the  photographic  process.  It  may  be  used 
however  in  pointing  the  instrument  to  the  desired  object. 
There  are,  however,  other  forms  of  reflecting  telescopes. 

49.  Simple  microscope. — The  word  microscope  usually 
means  an  instrument  used  for  magnifying  small  objects  close 
at  hand  which,  like  a  telescope,  has  two  optical  parts,  objective 
and  eyepiece:  but  in  stricter  language  such  an  instrument  is 
called  a  compound  microscope,  while  the  name  simple  micro- 
scope is  applied  to  a  single '  lens  used  as  a  magnifier  of  low 
power. 

A  simple  microscope  is  held  as  close  to  the  eye  as  con- 
venient, and  the  object  to  be  examined  is  placed  somewhat 
within  the  principal  focus,  so  that  the  eye  sees  a  magnified 
virtual  image  of  it  at  the  distance  which  is  most  suitable  for 
distinct  vision.  For  the  normal  eye  this  is  about  25cm.,  though 
it  differs  with  different  individuals.  Figure  57  shows  the  ar- 
rangement. L  is  the  magnifying  lens,  F  its  principal  focus, 


Figure  57 

AB  the  object,  A'B'  the  image  seen  by  the  eye.  We  take  as 
a  measure  of  the  magnifying  power  the  ratio  of  the  angle  which 
the  image  subtends  at  the  eye  to  that  which  the  object  itself 
would  subtend  if  the  lens  were  removed  and  the  object  put  back 
where  it  could  be  seen  most  distinctly,  that  is,  where  the  image 
is  in  the  figure.  Since  magnifying  powers  need  only  be  known 
roughly,  and  since  the  eye  is  placed  so  close  to  the  lens,  it  will 
suffice  to  consider  the  center  of  the  lens,  instead  of  the  pupil 


COMPOUND  MICROSCOPE  105 

of  the  eye,  as  the  apex  of  the  angles.     The  angle  subtended  by 
the  image  is 

A'B'/v  =  AB/u 
The  angle  which,  the  object  would  subtend  if  placed  at  the 

distance  v  is 

AB/v 
Therefore  the  magnifying  power  is 


AB/u  -f-  AB/v  = 


From  the  law  of  lenses 


I       I—I 
u"~     v~~£ 


--!=!- 
u  f 


-=!+!- 

u  f 

Therefore  the  magnifying  power  is  1  +  v/f  or  1  +  25/f  if  f 
is  expressed  in  centimeters.  As  an  example,  the  magnifying 
power  of  a  lens  of  5cm.  focal  length  is  6. 

A  Bamsden  eyepiece  used  as  a  single  lens  makes  a  very 
good  simple  microscope. 

50.  Compound  microscope. — It  is  not  practicable  to  get 
very  high  magnifying  power  with  a  single  lens,  for  that  would 
require  such  a  short  focal  length  that  the  lens  defects  such  as 
chromatic  and  spherical  aberration,  curvature  of  field,  etc., 
would  be  very  prominent.  Therefore,  wherever  high  magnify- 
ing power  is  necessary,  it  is  provided  by  a  compound  micro- 
scope. Figure  58  shows  the  arrangement  of  parts.  The  "ob- 
ject" is  usually  in  the  form  of  a  slide,  a  very  thin  slice  of  the 
material  to  be  examined  enclosed  between  two  thin  glass  plates. 
The  slide  is  represented  by  the  short  arrow  A  in  the  diagram. 
Slides  being  more  or  less  transparent,  they  are  examined  by 
transmitted  light".  A  mirror  M  and  a  condenser  C  concentrate 
upon  the  slide,  from  below,  a  beam  of  light  from  a  window  or 
other  broad  illuminated  area.  The  set  of  lenses  forming  the 
condenser  are  of  low  grade,  for  their  function  is  merely  to 
provide  illumination,  not  to  form  a  clear  image  of  anything. 


306 


LIGHT 


The  mirror  and  condensers  are  accessories  rather  than  parts  of 
the  microscope  proper.  The  latter  consists  of  an  objective  0 
and  an  eyepiece  FE.  The  former  is  shown  in  the  figure  as  a 


Figure  58 

single  hemispherical  lens,  but  it  is  in  fact  compounded  of 
several  distinct  units  in  order  to  secure  chromatic  and  other 
corrections.  The  eyepiece  shoAvn.  in  the  figure  is  what  is  called 
the  Huyghens  type,  which  has  certain  advantages  over  the 
Ramsden  eyepiece  described  in  connection  with  telescopes, 
though  it  has  also  the  disadvantage  that  it  cannot  be  used  in 
connection  with  crosshairs  or  micrometer.  If  it  is  necessary, 
as  sometimes  happens,  to  put  a  micrometer  on  a  microscope, 
the  Huyghens  eyepiece  must  be  exchanged  for  one  of  the 
Ramsden  type. 

The  objective  would,  but  for  the  eyepiece,  form  a  magni- 
fied real  inverted  image  of  the  slide  at  I15  but  the  field-lens  F 
intercepts  the  converging  light  directed  toward  this  image,  and 
converges  it  still  more,  forming  the  real  image  I2,  slightly 


PROJECTION  LANTERNS  10? 

smaller  and  slightly  lower.  Usually  a  diaphragm  is  placed  in 
the  plane  of  L  so  as  to  limit  the  visible  field  to  a  small  circle 
over  which  the  illumination  is  uniform.  The  light  diverging 
from  I,  then  passes  through  the  eye-lens  E,  which  forms  a  mag- 
nified virtual  image  at  I3,  the  image  which  the  eye  sees.  For 
very  high  powers,  an  "oil-immersion"  objective  is  used.  The 
objective  comes  very  close  to  the  slide,  and  the  space  between 
is  filled  with  a  drop  of  oil  having  an  index  of  refraction  nearly 
the  same  as  that  of  glass,  so  that  one  might  regard  the  object 
as  being  imbedded  in  the  objective.  Under  these  circumstances, 
the  resolving  power  of  the  microscope  is  somewhat  increased, 
and  the  brightness  is  also  increased  because  less  light  is  lost 
by  reflection  from  the  bottom  surface  of  the  objective  and  the 
top  surface  of  the  slide. 

Certain  particles  too  small  to  be  seen  with  the  bright  back- 
ground illumination  commonly  used  in  a  microscope,  such  as 
the  particles  in  certain  colloidal  solutions,  can  be  seen  as  bright 
points  against  a  dark  background  if  the  illumination  comes 
from  the  sides  instead  of  from  below.  This  is  the  principle  of 
the  so-called  ' '  ultramicroscope. ' ' 

In  the  figure,  the  fainter  lines  represent  the  full  beam  of 
light  from  the  point  of  the  arrow  through  the  objective  and 
eyepiece.  The  same  rays  are  shown  below,  from  before  they 
impinge  upon  the  mirror  till  they  strike  the  slide.  Similar 
rays  from  the  butt  of  the  arrow,  or  from  any  other  part  of  it, 
couid  be  drawn,  but  they  are  omitted  for  the  sake  of  simplicity. 


Figure   59 

51.  Projection  lanterns. — Figure  59  shows  the  ordinary 
projection  lantern,  for  throwing  upon  a  screen  an  enlarged 
image  of  a  lantern-slide.  The  slide  S  is  so  placed  with  refer- 
ence to  the  projecting  lens  L  that  the  screen  comes  at  the  con- 
jugate focus,  and  the  focal  length  of  L  must  be  chosen  with 
due  regard  to  the  distance  of  the  screen  and  the  desired  magni- 
fication. The  rest  of  the  apparatus  is  for  obtaining  the  neces- 
sary illumination  of  the  slide.  Since  the  latter  is  more  or  less 
trasparent,  the  illumination  is  supplied  by  transmitted  light. 


108  LIGHT 

p  is  the  positive,  n  the  negative  carbon  of  an  arc-lamp,  and  C 
a  condenser  consisting  of  two  plano-convex  rough  lenses.  The 
slide  should  lie  as  close  to  the  condenser  as  convenient,  and 
the  arc  should  be  so  placed  that  its  light  is  focussed  by  the 
condenser  through  the  slide  upon  the  center  of  the  projection 
lens,  forming  an  image  of  the  arc  there.  The  lens  L  should 
be  achromatic,  in  order  to  avoid  chromatic  aberration  at  the 
screen;  and  it  should  consist  of  two  units  with  a  diaphragm 
between,  otherwise  the  picture  on  the  screen  may  show  some 
distortion.  Except  for  these  two  points  L  need  not  be  a  high- 
grade  lens.  The  pencil  of  light  coming  from  any  point  on  the 
slide  is  so  narrow,  as  shown  in  the  figure,  that  there  is  little 
opportunity  for  spherical  aberration,  astigmatism,  or  curva- 
ture of  field  to  show  any  bad  effect. 

The  focussing  is  done  by  moving  the  lens  L.  If  the  illumi- 
nation of  the  image  is  not  uniform,  this  is  an  indication  that 
the  arc  is  not  in  the  proper  place,  or  that  the  negative  carbon 
is  shutting  off  some  of  the  light  from  the  positive  carbon. 
Lately  it  has  become  more  common  to  substitute  a  high-power 
filament  lamp  instead  of  the  arc.  This  makes  the  lantern  much 
easier  to  operate,  and  the  illumination,  though  slightly  weaker, 
is  strong  enough  in  most  cases. 

The  opaque  projec- 
tion lantern,  one  form  of 
which  is  shown  in  figure 
60,  is  used  for  projecting 
images  of  postcards,  pic- 
tures or  printed  matter  in 
books,  etc.  It  differs  from 
the  slide-lantern  in  two  es- 
6u  sential  points.  First,  it  is 

necessary  to  introduce  a  reflection  (mirror  m  in  the  figure)  be- 
tween picture  and  screen,  to  prevent  the  image  from  being  either 
upside  down  or  right  side  to  left.  Second,  the  original  picture 
must  be  illuminated  from  the  front,  since  transmitted  light  is  out 
of  the  question,  and  this  makes  it  difficult  to  get  the  illumina- 
tion strong  enough.  The  projection  lens  L  must  be  of  excep- 
tionally large  diameter  for  its  focal  length,  and  it  must  be  well 
corrected  for  all  the  defects  of  lenses,  since  it  receives  a  full 
beam  of  light.  The  arc  is  made  to  give  an  exceptionally  large 


PROJECTION  LANTERNS  100 

amount  of  light  by  using  large  carbons  and  a  very  heavy  cur- 
rent, and  the  condenser  system  is  made  to  cover  a  very  large 
angle  from  the  arc.  Usually  a  glass  cell  containing  water  is 
interposed,  to  cut  out  much  of  the  infrared  light,  which  would 
unduly  heat  the  picture.  For  use  in  small  rooms,  where  the 
picture  on  the  screen  need  not  be  very  large,  a  high-power 
tungsten  filament  lamp  may  replace  the  arc,  but  in  such  a 
case  it  is  necessary  to  use  a  special  screen  made  of  filled  can- 
vas covered  with  aluminum  paint,  which  reflects  more  strongly 
than  a  simple  white  screen. 

Problems. 

1.  The  objective  of  a  telescope  has  a  focal  length  of  30 
ft.    "What  is  the  magnifying  power,  when  an  eyepiece  of  focal 
length  y2  inch  is  used? 

2.  Explain  why  it  is  that  dirt,  or  even  a  large  opaque 
obstacle,  on  the  surface  of  the  objective  of  a  telescope,  is  never 
visible  to  a  person  looking    through;    the    eyepiece,    the    only 
apparent  effect  being  a  general  dimming  of  the  image. 

3.  Prove  that  the  "  exit-pupil "  is  the  image  of  the  objec- 
tive as  formed  by  the  eyepiece. 

4.  A   projection  lantern  is  being  planned  for  use   in   a 
certain  room.     The  screen  is  to  be  30  ft.  from  the  slide,  and 
it  is  desired  that  the  image  of  the  slide  on  the  screen  shall 
measure  58.5  X  72  inches.     (A  slide  is  3.25  X  4  inches).    What 
must  be  the  focal  length  of  the  projection  lensf 

5.  Explain  completely  why  strong  illumination  is  so  much 
harder  to  obtain  with  the  opaque  projection  lantern  than  with 
the  ordinary  lantern  for  slides. 


CHAPTER  VII. 

52.  Prism  spectroscope. — 53.  Bright-line  spectra. — 54.  Spectral 
series. — 55.  Continuous  spectra. — 56.  Dark-line  spectra. — 57.  Absorp- 
tion by  solids  and  liquids. — 58.  Continuous  spectrum  of  an  absolutely 
black  body. — 59.  Planck's  theory  of  "quanta." — 60.  The  plane  grating. 
— 61.  Why  the  lines  are  sharp. — 62.  Reflection  gratings. — 63.  The  con- 
cave grating. — 64.  The  ultraviolet  region.  Fluorescence.  Phosphor- 
escence. Photography. — 65.  The  infrared  region. — 66.  The  bolometer. 
— 67.  The  thermopile. — 68.  The  Doppler  principle.  Motion  of  the  stars. 

52.  Prism  spectroscope. — We  have  already  considered  the 
spectroscope  to  some  extent,  in  the  chapter  on  color,  but  we  are 
now  in  a  better  position  to  understand  its  principles.  The 
essential  parts  of  a  prism  spectroscope,  shown  in  figure  61,  are 
the  collimator  C,  the  prism  P  (or  a  train  of  prisms),  and  the 

telescope  T.  The  colli- 
mator is  a  tube,  at 
one  end  of  which  is 
an  achromatic  lens,  at 
the  other  end  a  fine 
slit.  The  latter  is 
carefully  adjusted  at 
the  principal  focus  of 
the  former,  so  that 

light  which  enters  the  slit  and  passes  through  the  lens  emerges 
in  accurately  plane  waves.  The  beam  then  passes  through  the 
prism  and  is  dispersed,  that  is,  the  different  wavelengths  are 
deviated  to  different  amounts.  It  is  best  to  have  the  prism 
turned  so  that  the  region  of  the  spectrum  to  be  examined 
traverses  it  at  minimum  deviation. 

It  should  be  borne  in  mind  that  when  the  light  leaves  the 
prism;  although  the  different  wavelengths  take  different  direc- 
tions, yet  all  the  rays  of  -any  one  wavelength  remain  parallel 
to  one  another  till  they  strike  the  objective  of  the  telescope. 
Therefore  the  latter  converges  the  waves  of  each  particular 
length  to  a  definite  place  in  the  principal  focal  plane.  Con- 
sequently, there  will  be  in  this  plane  an  image  of  the  slit  for 
each  particular  wavelength  that  enters  the  slit,  and  the  whole 

(110) 


BRIGHT-MNE  SPECTRA  111 

array  of  these  images  constitute  the  spectrum  of  the  light  in 
question.  The  eyepiece  of  the  telescope  forms  an  enlarged  vir- 
tual image  of  the  spectrum,  just  as  it  would  form  a  virtual 
image  of  any  very  distant  object  which  the  objective  focussed 
in  its  focal  plane. 

A  spectrometer  is  a  spectroscope  provided  with  a  large 
divided  circle,  so  that  the  angular  position  of  the  prism,  of  the 
telescope,  or  of  both,  can  be  accurately  measured.  There  is 
also  a  crosshair  at  the  principal  focus  of  the  telescope.  The 
spectrometer  is  used  for  making  accurate  measurements  of  the 
refractive  indices  of  prisms,  and  for  other  angular  measure- 
ments. 

A  spectrograph  is  a  spectroscope  so  arranged  that  the 
spectrum  can  be  photographed  instead  of  viewed  directly.  Any 
spectroscope  can  be  converted  into  a  spectrograph  by  removing 
the  eyepiece  from  the  telescope  and  putting  a  photographic 
plate  in  the  focal  plane  of  the  objective;  but  it  is  better  to 
remove  the  whole  telescope,  replacing  it  by  what  is  really  a 
long-focus  camera,  a  light- tight  box  with  a  specially  corrected 
photographic  lens  at  one  end  and  a  holder  for  photographic 
plates  at  the  other.  An  ordinary  hand  camera,  focussed  for 
distant  objects,  might  be  used  instead,  but  in  most  cameras  the 
focal  length  is  rather  short,  and  this  causes  the  spectrum  also 
to  be  short,  since  its  length  is  proportional  to  the  focal  length 
of  the  projecting  lens,  other  things  being  equal. 

53.  Bright-line  spectra. — The  character  of  the  spectrum 
seen  in  a  spectroscope  varies  greatly  with  the  chemical  nature 
and  physical  condition  of  the  body  emitting  the  light  that 
passes  in  through  the  slit.  The  flame  of  a  Bunsen  burner, 
except  for  the  small  bluish  inner  cone,  is  practically  invisible, 
and  if  such  a  flame  is  placed  before  the  slit,  nothing  is  seen  on 
looking  into  the  spectroscope,  as  we  should  expect.  But  if  a 
piece  of  asbestos  soaked  in  a  solution  of  common  salt  (sodium 
chloride,  NaCl)  or  of  any  other  compound  of  sodium,  is  put 
into  the  edge  of  the  flame,  the  latter  immediately  becomes  yel- 
low in  color.  If  this  yellow  light  enters  the  slit,  the  spectrum 
shows  two  fine  yellow  lines,  that  is  two  yellow  images  of  the 
slit.  The  light  has  a  slightly  different  wavelength  in  these 


112  LIGHT 

images,  .00005890cm.  in  one  and  .00005896cm.  in  the  other.* 
We  interpret  the  appearance  of  these  lines  as  follows:  atoms 
of  sodium  pass  into  the  flame  and,  under  the  conditions  exist- 
ing there,  start  into  vibration  with  two  different  periods,  thus 
starting  in  the  ether  waves  of  the  two  different  wavelengths 
given. 

The  fact  that  these  particular  lines  appear  in  the  spectrum, 
whatever  compound  of  sodium  be  used,  proves  that  it  is  the 
sodium,  not  the  chlorine,  that  is  responsible  for  them.  Chlorine 
produces  no  color  in  a  flame,  though  it  can  be  excited  to  radia- 
tion by  an  electric  spark.  The  appearance,  in  the  spectrum  of 
any  source  of  light,  of  two  bright  lines  of  wavelength  .00005890 
cm.  and  .00005896  cm.  is  therefore  a  sure  and  delicate  test  for 
the  presence  of  sodium.  The  merest  traces  of  sodium,  far  too 
slight  for  detection  by  chemical  methods,  produce  the  distinc- 
tive coloration  in  a  flame.  One  cannot,  however,  assume  that 
a  yellow  color  alone  indicates  sodium,  for  yellow  includes  a 
considerable  range  of  wavelengths,  and  some  other  elements 
have  in  their  spectra  yellow  lines,  of  wavelength  different  from 
those  attributable  to  sodium.  For  instance,  if  the  light  from 
a  Cooper-Hewitt  electric  lamp  (mercury  arc)  be  examined  with, 
a  spectroscope,  it  shows  a  number  of  lines,  two  of  which  have 
wavelengths  .00005769cm.  and  .00005790cm.,  which  brings  them 
in  the  yellow  region,  but  a  slightly  different  part  of  the  yellow 
from  the  sodium  lines. 

In  the  Bunsen  flame,  sodium  never  shows  anything  but 
the  two  above-mentioned  lines  and  a  very  faint  green  one,  but 
there  are  circumstances  in  which  it  gives  in  addition  a  number 
of  other  lines;  as  when  metallic  sodium  or  a  salt  of  sodium  is 
put  into  the  crater  of  a  carbon  arc-light. 

*The  two  lines  are  so  nearly  the  same  in  wavelength  that  when  a 
single  small  prism  is  used  in  the  spectroscope  they  appear  as  a  single 
line.  Spectroscopes  of  higher  power  show  them  as  separate  and  dis- 
tinct, and  the  most  powerful  even  show  that  the  wavelength  is  not 
absolutely  definite  in  either  line.  Each  line  has  a  small  but  perceptible 
ividth,  showing  that  for  each  the  wavelength  varies  between  certain 
narrow  limits.  This  statement  is  also  true  of  all  other  spectrum  lines, 
and  it  is  impossible  to  obtain  a  bean)  of  light  all  of  which  has  exactly 
the  same  wavelength. 


BRIGHT-LINE  SPECTRA  113 

In  order  to  study  the  spectra  oi  the  different  elements, 
various  means  must  be  employed  to  get  them  into  a  condition 
where  they  emit  their  characteristic  radiations.  There  are 
only  a  few  elements  which  give  their  spectra  in  a  flame,  like 
sodium.  An  effective  method  in  the  case  of  metallic  elements 
which  do  not  rapidly  oxidize  or  suffer  other  chemical  change 
in  air,  is  to  pass  an  electric  spark  between  points  made  from 
the  metal  in  question,  the  spark  being  operated  by  an  induction 
coil  or  transformer  in  parallel  with  a  condenser.  Another 
method  is  to  take  two  carbon  rods,  bore  a  hole  in  one  of  them, 
fill  it  with  the  metal,  and  connect  both  rods  to  the  terminals 
of  a  direct-current  supply  of  not  too  low  voltage  with  some 
resistance  in  series.  When  the  ends  of  the  rods  are  touched 
together  and  separated  about  a  quarter  of  an  inch,  the  intense 
heating  at  the  point  of  contact  vaporizes  the  carbon  and  forms 
a  bridge  of  glowing  vapor  called  the  arc,  across  which  the  cur- 
rent continues  to  flow.  Some  of  the  element  packed  into  the 
hole  in  the  rod  also  vaporizes  and  contributes  its  vapor  to  the 
formation  of  the  arc.  If  the  light  from  the  arc  itself  (the 
bridge  of  vapor,  not  the  glowing  ends  of  the  rods)  is  passed 
through  the  slit  of  the  spectroscope,  a  large  number  of  lines 
appear,  some  of  which  are  due.  to  the  element  in  question,  some 
to  gaseous  compounds  of  carbon,  and  some  to  such  impurities 
as  are  always  present  in  the  rods. 

The  spectrum  of  a  gas  is  usually  obtained  by  the  use  of  a 
so-called  vacuum-tube.  This  is  a  glass  tube  with  a  restricted 
middle  portion,  into  opposite  ends  of  which  are  sealed  metallic 
terminals.  The  tube  is  evacuated  of  air,  and  enough  of  the 
gas  is  put  in  to  exert  a  pressure  of  a  few  millimeters  of  mer- 
cury, after  which  the  tube  is  hermetically  sealed.  The  electric 
discharge  of  an  induction  coil  is  sent  through  the  tube,  from 
one  terminal  to  the  other,  causing  the  gas  inside  to  become 
luminous  and  emit  its  characteristic  wavelengths. 

Every  known  element  has  more  than  one  line  in  its  spec- 
trum. No  two  elements  have  identical  spectra,  and  so  far  as  is 
known  no  two  elements  show  the  same  line  in  common,  with, 
the  possible  exception  of  hydrogen  and  helium. 

Besides  the  elements,  certain  compounds  also  emit  spectra 
composed  of  lines.  Thus,  the  blue  inner  cone  of  a  Bunsen  burner 
shows  the  spectrum  of  carbon  monoxide,  and  there  are  several 


114  LIGHT 

groups  of  lines  in  the  spectrum  from  the  carbon  arc  which  are 
believed  to  originate  in  cyanogen  gas,  produced  by  the  action 
of  atmospheric  nitrogen  on  the  carbon  poles.  The  lines  in  the 
spectra  of  compounds  are  arranged  in  groups  of  more  or  less 
regular  order,  technically  known  as  bands. 

54  Spectral  series. — The  fact  that  a  single  element  emits 
light  of  several  wavelengths  shows  that  the  atom  is  capable  of 
vibrating  in  several  different  frequencies,  just  as  a  stretched 
string  or  column  of  air  can  execute  the  vibrations  that  produce 
sound  waves  in  several  definite  frequencies.  A  string  can  vibrate 
not  only  with  its  fundamental  frequency,  which  we  may  call  N, 
giving  a  sound  of  wavelength  L,  but  also  with  a  frequency 
twice  as  high,  2N  (the  octave),  giving  a  wavelength  L/2,  a 
frequency  3N,  giving  wavelength  L/3,  and  so  on  indefinitely. 
Therefore  we  might  represent  the  whole  series  of  sound  wave- 
lengths emitted  by  the  string  with  the  single  formula 

L 

A  =  — 
n 

where  L  is  a  certain  constant  for  the  string,  and  n  may  have 
any  integral  value  from  1  to  oo.  When  n  =  1,  A  is  the  wave- 
length of  the  fundamental,  when  n  =  2,  A  is  the  wavelength 
of  the  first  overtone,  etc. 

A  most  natural  question  is  the  following :  Are  not  also 
the  different  wavelengths  of  light,  given  out  by  such  an  ele- 
ment as  sodium  or  hydrogen,  related  to  one  another  in  some 
simple  numerical  way,  so  that  a  single  formula  will  represent 
all  of  them  if  different  integral  values  are  given  to  one  of  the 
symbols?  This  cannot  be  answered  by  an  unqualified  yes  or 
no,  but  we  can  say  that  in  some  of  the  elements  (particularly 
the  metallic  ones  of  small  atomic  weight)  some  of  the  lines  can 
be  represented  in  this  way,  though  not  by  so  simple  a  formula 
as  applies  to  the  acoustical  vibrations  of  a  string.  The  simplest 
case  is  that  of  hydrogen.  Figure  62  is  a  photograph  of  the 
visible,  and  part  of  the  ultraviolet,  regions  in  the  spectrum  of 
this  gas.  A  careful  examination  shows  that  the  lines  may  be 
classified  into  two  groups-  First,  there  is  a  large  number  of 
lines  without  any  apparent  regularity  of  arrangement  what- 
ever. Second,  there  are  several  lines,  marked  a,  /?,  y,  etc.,  on 


SPECTRAL   SERIES  115 

the  photograph,  which  show  regularity  in  two  respects.  The 
one  of  greatest  wavelength,  a,  is  the  strongest  line  in  the 
whole  spectrum,  and  each  succeeding  line  of  the  group  is 
weaker  than  the  one  before  it-  Moreover,  when  the  wavelengths 


Figure  62 

of  these  lines  are  measured,  it  is  found  that  in  passing  through 
the  series,  from  a  through  (3,  y,  8,  etc.,  they  come  closer  and 
closer  in  wavelength,  like  a  mathematical  series  approaching  a 
limit.  Balmer  showed  that  the  wavelengths  of  the  whole  group, 
including  lines  too  far  in  the  ultraviolet  to  show  in  this  photo- 
graph, can  be  represented  with  a  considerable  degree  of  accura- 
cy by  the  single  formula 

A.  =. 00003646  Q2^4 

where  n  may  take  any  integral  value  from  3  on.  If  n  =  3, 
we  get  the  wavelength  of  the  a  line, — if  n  =  4.  we  get  /?, — 
and  so  on*.  This  whole  group  of  lines,  represented  by  a 
single  formula,  constitutes  a  spectral  series.  Our  knowledge 
of  series  has  been  greatly  increased  by  the  work  of  Rydberg, 
Kayser  and  Runge,  and  many  other  investigators.  The  hydro- 
gen spectrum  shows  another  series  in  the  far  ultraviolet  region, 
and  still  another  in  the  infrared.  Other  elements  also  show 
series  in  their  spectra,  although  a  slightly  more  complicated 
formula  is  necessary  to  represent  them.  A  comprehensive  ac- 
count of  series  in  spectra  is  found  in  French  in  the  "  Rapports 
Presentes  au  Congres  International  de  Physique,"  1900.  A 
very  good  resume  in  English  is  contained  in  pages  559  to  621 
of  Baly's  ' '  Spectroscopy, "  second  edition.  In  this  chapter  we 
shall  not  take  up  the  attempts  to  explain  the  peculiar  form  of 

*The  measured  wavelengths  of  the  first  four  of  these  lines,  in 
centimeters,  are  .00006563,  .00004862,  00004341,  .00004102. 


116  LIGHT 

the  series  relation,  since  the  subject  conies  more  appropriately 
after  the  introduction  of  the  electromagnetic  theory  of  light. 

55.  Continuous  spectra. — Spectral  lines  tend  to  become 
widened  when  the  density  of  the  radiating  gas  or  vapor  is  in- 
creased. The  hydrogen  lines  produced  when  a  spark  is  passed 
between  platinum  points  in  the  gas  at  atmospheric  pressure  are 
much  broader  and  hazier  than  when  it  is  in  the  rarefied  con- 
dition of  the  vacuum-tube.  If  a  large  amount  of  sodium  is 
present  in  the  carbon  arc,  the  two  strong  lines  in  the  yellow, 
instead  of  being  fine  and  sharp,  become  very  broad,  and  can 
easily  be  made  to  run  together  and  extend  some  distance  be- 
yond their  original  positions  on  both  sides,  causing  them  to 
have  the  appearance  of  a  single  broad  yellow  band  with  hazy 
edges.  This  of  course  means  that  the  wavelengths  emitted  are 
no  longer  confined  even  approximately  to»  two  definite  numeri- 
cal values,  but  extend  over  a  relatively  wide  range.  Several 
causes  contribute  to  this  effect  of  increased  density,  one  of 
which  is  probably  the  fact  that  any  given  atom  is  hindered,  by 
the  very  close  proximity  of  other  atoms,  in  its  natural  free 
vibrations.  At  any  rate,  when  an  exceedingly  dense  gas,  or  a 
solid  or  liquid  body,  becomes  luminous,  the  widening  of  the 
characteristic  lines  is  so  extreme  that  all  possible  wavelengths 
are  emitted  within  the  range  of  the  visible  spectrum  and  be- 
yond, and  all  appearance  of  definite  lines  is  lost.  The  spec- 
trum is  then  said  to  be  continuous,  since  it  extends  throughout 
a  very  wide  range  of  wavelengths  without  a  break  in-  continuity 
anywhere.  In  contradistinction  from  this,  the  kind  of  spectrum 
that  we  have  found  to  be  given  out  by  rare  gases,  metallic 
vapors,  etc.,  is  called  a  bright-line  spectrum.  As  an  example, 
the  hot  carbon  pole  of  an  arc  light  gives  a  continuous  spectrum, 
but  the  bridge  of  vapor  between  the  two  poles  gives  a  bright- 
line  spectrum. 

The  following  application  of  the  principles  of  the  spectro- 
scope to  astronomical  problems  is  interesting  and  instructive. 
A  nebula  is  a  celestial  object  which  appears  in  the  telescope  as 
a  cloud  of  gas,  but  the  possibility  exists  that  it  may  really  be 
a  swarm  of  stars,  so  close  together  and  so  far  from  us  that  the 
telescope  is  incapable  of  resolving  them  into  discrete  bodies. 
The  spectroscope,  however,  shows  that  some  nebulae  have  a  con- 


DARK-LINE  SPECTRA  117 

tinuous  spectrum,  others  a  bright-line  spectrum,  so  that  with 
the  aid  of  this  instrument  it  is  easy  to  pick  out  those  that  are 
gaseous. 

56.  Dark-line  spectra.— The  sun  and  most  of  the  stars 
show  still  a  third  type  of  spectrum,  which  may  be  said  to  be 
an  exact  reversal  of  the  bright-line  type.  While  the  latter  is 
an  assemblage  of  scattered  bright  lines,  in  colors  appropriate 
to  the  spectral  region  in  which  they  fall,  against  a  blank — that 
is  a  black — background,  the  former  is  an  assemblage  of  fine  black 
lines  against  an  otherwise  continuous  colored  background.  It 
is  therefore  called  a  dark-line  spectrum.  It  may  be  described  as 
a  continuous  spectrum  with  certain  definite  wavelengths .  miss- 
ing. 

The  following  experiment  explains  the  cause  of  these  dark 
lines  in  the  solar  spectrum.  An  arc  lamp,  a  Bunsen  burner,  a 
converging  lens,  and  a  spectroscope  are  set  up  in  line,  so  that 
the  lens  forms  an  image  of  the  bright  carbon  pole  on  the  slit 
of  the  spectroscope,  and  the  light  passes  through  the  flame  of 
the  burner  before  reaching  the  slit.  Some  of  the  light  passes 
into  the  collimator,  and  of  course  produces  a  continuous  spec- 
trum. A  little  common  salt  is  inserted  in  the  flame,  and  im- 
mediately the  sodium  lines  appear,  not  bright,  however,  as 
they  would  be  in  the  absence  of  the  light  from  the  arc,  but  as 
apparently  dark  lines  against  the  bright  continuous  spectrum 
of  the  arc.  The  sodium  vapor  absorbs  from  the  light  passing 
through  it  those  particular  wavelengths  which,  it  is  capable  of 
emitting,  and -absorbs  more  than  it  emits,  thus  making  the  lines 
appear  black  against  the  brighter  arc  spectrum,  though  in 
reality  they  are  not  absolutely  black.  Undoubtedly,  the  dark 
lines  of  the  sun's  spectrum  are  produced  by  absorption  in  the 
same  way.  The  main  bulk  of  the  sun  is  believed  to  be  an 
exceedingly  dense  gaseous  mixture,  as  viscous  as  a  liquid,  and 
like  a  hot  liquid  it  gives  a  strictly  continuous  spectrum.  But 
surrounding  this  dense  luminous  portion  is  an  envelope  of 
cooler  and  rarer  vapor  containing  many  chemical  elements. 
These  absorb  from  the  light  passing'  through  them  just  those 
wavelengths  which  they  can  emit.  When  the  light  reaches  the 
earth  it  is  therefore  deprived  of  these  particular  wavelengths, 
and  the  spectroscope  shows  black  lines  at  the  corresponding 


118  LIGHT 

positions  in  the  spectrum.  By  comparing-  the  positions  of  the 
black  lines  with  the  positions  of  the  bright  lines  emitted  by 
various  terrestrial  elements,  it  has  been  possible  to  identify  on 
the  sun  most  of  the  elements  known  to  us  on  the  earth.  The 
moon  and  the  planets,  since  they  send  us  only  light  which  they 
receive  from  the  sun,  have  the  same  spectrum.  Most  of  the 
fixed  stars  have  spectra  of  the  same  character  as  the  sun's,  and 
in  some  cases  only  a  careful  examination  can  distinguish  them 
from  the  latter.  Thus  the  spectroscope  proves  that  the  fixed 
stars  are  bodies  of  the  same  general  physical  condition  as  our 
sun,  in  spite  of  great  differences  in  size,  mass,  and  temperature; 
and  it  also  shows  that  the  chemical  elements  present  in  the 
earth  are  distributed  throughout  the  universe  and  probably 
make  up  the  major  part  of  its  material,  a  fact  which  could 
hardly  have  been  proved  by  any  other  means. 

57.  Absorption  by  solids  and  liquids. — Solids  and  liquids 
also  produce  absorption  of  light,  as  we  have  already  seen  in 
the  chapter  on  color.  If  a  colored  liquid,  such  as  a  solution  of 
copper  sulphate,  potassium  permanganate,  or  chlorophyll,  is 
placed  in  front  of  the  slit  of  a  spectroscope,  and  the  light  from 
a  source  that  would  of  itself  alone  give  a  continuous  spectrum 
is  passed  through  it,  certain  parts  of  the  spectrum  are  absorbed 
in  whole  or  in  part.  The  black,  or  darkened,  regions  which 
then  appear  in  the  spectrum  are  not  fine  and  sharp,  like  those 


figure  63 

produced  by  sodium  vapor  in  a  Bunsen  flame,  but  broad  and 
hazy.  They  are  called  absorption  bands.  Figure  63  (A)  is  a 
photograph  of  the  absorption  spectrum  of  an  alcoholic  solution 
of  chlorophyll,  the  green  coloring  matter  of  plants.  Figure  63 
(B)  is  a  photograph  of  the  complete  spectrum,  with  the  chloro- 
phyll absent,  to  serve  as  a  comparison. 


BLACK-BODY  SPECTRUM 


119 


58.  Continuous  spectrum  of  an  absolutely  black  body. — 
The  strictly  continuous  spectra  given  out  by  hot  solids  and 
liquids  differ  from  one  another  only  as  regards  the  distribu- 
tion of  the  energy  in  different  wavelengths.  Thus,  the  spec- 
trum from  the  pole  of  a  carbon  arc  is  not  only  brighter  through- 
out than  that  from  the  filament  of  a  tungsten  incandescent 
lamp, — it  also  has  a  greater  proportio.n  of  its  energy  in;  the 
shorter  wavelengths  than  that  from  the  same  body  when  cool- 
er; but  differences  in  material  and  surface  condition  have  also 
great  effect.  The  only  kind  of  continuous  spectrum  that  can 
be  theoretically  studied  is  that  from  what  is  called  an  absolute - 


Figure  64 

ly  black  body,  which  is  defined  as  one  which  absorbs  all  light, 
of  whatsoever  wavelength,  that  falls  upon  it,  reflecting1  none. 
Strictly  speaking  no  objects  are  absolutely  black,  though  such 
materials  as  lampblack  and  platinum-black  nearly  fulfill  the 
definition.  But  it  can  be  shown  by  theory  that  the  inside  of 
an  enclosure,  the  walls  of  which  are  kept  at  a  uniform  and 
constant  temperature,  acts  exactly  like  a  theoretical  absolutely 
black  body;  and  it  is  possible  to  make  use  of  this  fact  for 
experimental  purposes  by  making  a  small  hole  in  the  wall  of 
such  an  enclosure,  through  which  light  from  the  interior  can 
pass  out  and  enter  the  slit  of  a  spectroscope.  Figure  64  is  a 
series  of  graphs,  drawn  for  different  temperatures  of  the 
radiating  enclosure,  of  the  results  of  measurements  of  such 
black-body  continuous  spectra.  For  each  point  on  a  curve,  the 
abscissa  represents  the  wavelength,  the  ordinate  the  correspond- 
ing energy.  It  will  be  noticed  that  for  higher  temperatures 


120  LIGHT 

the  energy  is  greater  throughout,  but  particularly  in  the  short- 
er wavelengths. 

59.  Planck's  theory  of  "quanta".— The  relation  between 
the  energy  of  black  body  radiation  and  the  wavelength,  for  a 
given  temperature,  as  represented  by  the  curves  of  figure  64, 
has  been  the  subject  of  many  exhaustive  theoretical  investiga- 
tions. It  is  extremely  difficult  to  give  a  complete  theoretical 
explanation  of  the  exact  form  of  these  curves.  Indeed,  the 
man  who  made  most  progress  toward  this  end,  Max  Planck, 
came  to  the  conclusion  that  an  explanation  is  impossible  unless 
we  make  a  very  remarkable  hypothesis  in  regard  to  the  be« 
havior  of  a  radiating  atom,  which  amounts  to  this — that  al- 
though an  atom  can  absorb  energy  steadily  and  continuously, 
it  can  radiate  only  if  and  when  it  has  acquired  by  absorption 
a  certain  definite  quantum  of  energy,  or  an  integral  number 
of  times  that  quantum;  and  when  it  does  radiate  it  radiates 
away  all  the  energy  it  contains.  Just  as  by  adopting  the  atomic 
theory  of  matter  we  abandon  the  ancient  notion  that  matter 
is  continuous,  so  Planck's  hypothesis  would  lead  to  the  con- 
clusion that  energy  also,  so  far  as  the  radiation  of  it  is  con- 
cerned, is  composed  of  discrete  amounts.  For  an  atom  can 
radiate  one  quantum,  or  two,  or  three,  etc.,  but  not  one  and  a 
fraction. 

This  hypothesis,  which  has  been  named  the  "quantum 
theory,"  is  so  very  different  from  our  previous  notions  about 
energy  that,  in  spite  of  Planck's  success  in  deriving  a  formula 
for  black  body  radiation  which  fits  the  experimental  curves,  it 
is  doubtful  if  it  could  obtain  much  support  were  it  not  that  a 
number  of  other  phenomena,  including  such  diverse  things  as 
the  variation  of  specific  heats  with  temperature,  X-rays,  and 
the  explanation  of  spectral  series,  are  made  much  more  under- 
standable by  means  of  the  same  hypothesis.  Planck  offers  no 
explanation  of  why  an  atom  should  radiate  in  such  a  manner, 
and  the  whole  question  of  the  quantum  theory  is  one  of  the 
puzzles  of  modern  physics. 

The  size  of  the  quantum  is  not  the  same  for  all  wave- 
lengths, but  is  directly  proportional  to  the  frequency,  or  in- 
versely to  wavelength.  That  is,  for  any  frequency  v  the  small- 
est unit  of  energy  radiated  is 


PLANE  GRATING 


121 


The  multiplier  h  is  an  absolute  constant,  whose  numerical  value, 
in  the  c.  g.  s.  system  of  units  is 

h  =  6.55  X  10-27 
It  is  commonly  known  as  "Planck's  constant." 

60.  The  plane  grating. — So  far,  we  have  learned  no  way  of 
measuring  wavelengths  except  by  simple  interference  experi- 
ments, such  as  that  of  Fresnel  with  the  two  mirrors,  as  described 
in  section  22.  That  such  a  method  is  not  capable  of  much,  ac- 
curacy can  be  seen  from  the  following  considerations.  Refer- 
ring again  to  figure  20,  section  22,  it  will  be  recalled  that  we 
found  that  certain  points  C,  M^,  M/,  M2,  M/,  etc.,  are  very 
bright,  and  the  points  midway  between  dark.  The  determination 
of  wavelength  is  made  by  measuring  the  interval  between  two 
successive  bright  spots,  and  also  the  distance  between  the  two 
sources  ST  and  S2,  and  their  distance  from  the  plane  of  the 
screen.  Not  only  is  the  distance  SjS,,  difficult  to  measure,  but 
also  the  distance  between  the  bands,  CMi,,  M^IVL,  M2M3,  etc., 
cannot  be  measured  accurately,  because  the  bright  points  are 
not  sharply  defined  but  shade  off  gradually  into  darkness. 

The  difficulty  might  be  illustrated  graphically  by  plotting 
abscissas  to  the  right  of  the  line  AB  in  figure  20,  the  length  of 
each  abscissa  representing  the  intensity  of  the  illumination  at 
the  corresponding  point  on  the  screen.  The  graph  that  would 
be  obtained  would  be  like  fig- 
ure £5  (X).  Evidently  the  loca- 
tion of  the  places  of  maximum 
brilliancy  is  subject  to  consid- 
erable error,  which  would  be 
much  lessened  if,  instead  of 
these  broad  maxima,  we  had 
sharp  and  clearcut  bright  lines 
separated  by  broad  dark  spaces, 
as  indicated  in  figure  65  (Y).  x  Y 

Two   other   decided   advantage?  Figure  6S 

would  also  accrue :  first,  the  maxima  would  be  much  brighter, 
since  the  light  would  be  confined  to  a  very  narrow  instead  of 
a  broad  band, — second,  if  more  than  one  wavelength  were  pres- 
ent, the  maxima  due  to  the  different  colors  would  be  much  less 
likely  to  overlap. 


122 


LIGHT 


The  desired  result,  making  the  maxima  narrow,  sharp  and 
bright,  can  be  secured  by  getting  interference  from  more  than 
two  points  at  once.  For  instance,  if  Fresnel's  mirror  experi- 
ment could  be  arranged  so  that  there  were  three  regularly 
spaced  apparent  sources  of  light,  instead  of  only  the  two,  S: 
and  S2,  the  maxima  would  be  sharper  and  brighter, — if  there 
were  four,  they  would  be  still  sharper, — and  so  on;  but  one 
of  the  best  devices  for  the  purpose  is  what  is  called  a  grating. 

In  its  theoretically  simplest  form,  a  grating  is  an  opaque 
plate  containing  a  large  number  of  slits,  parallel  and  spaced 
olose  together  at  equal  intervals.  Light  from  a  narrow  source, 
like  a  distant  slit,  or  a  star,  falls  upon  it  and  passes  through 
the  many  narrow  slits,  producing  interference  bands  on  the 
other1  side.  Let  AB,  figure  66,  represent  a  section  of  such  a 


Figure  66 

grating.  The  slits,  shown  in  section  at  a,  b,  c,  etc.,  are  sup- 
posed to  run  perpendicular  to  the  plane  of  the  paper.  We  shall 
suppose  that  monochromatic  plane  waves  are  falling  perpendic- 
ularly upon  it,  the  advancing  wavefronts  being  indicated  at 
C.  The  slits,  or  transparent  portions  of  the  grating,  may  be 
regarded  as  centers  froni  which  new  wavelets  start  out.  As 
these  get  farther  from  the  grating,  where  their  curvature  is 
less,  they1  tend  to  combine  into  several  different  sets  of  plane 
wavefronts,  moving  in  different  directions.  For  instance,  the 
12th  wave  out  from  a,  together  with  the  12th  from  each  of  the 
other  openings,  tends  to  form  a  plane  wave,  parallel  to  the 


PLANE  CRATING  123 

incident  waves  C.  There  will  ber  a  continuous  train  of  such 
waves,  a  few  of  which  are  indicated  farther  out  at  X.  These, 
after  passing  through  the  lens,  will,  be  brought  to  the  prin- 
cipal focus  P,  and  will  form  a  bright  spot  there,  exactly  as  if 
the  grating  were  removed  and  the  original  wavefronts  C  fell 
directly  upon  the  lens.  An  entirely  different  set  of  wavefronts 
will  be  formed  by  a  combination  of  the  12th  wavelet  out  from 
a,  the  llth  from  b,  the  10th  from  c,  and  so  on,  the  resulting 
wavefront  being  inclined  at  a  certain  angle  to  the  original 
wavefronts.  A  few  of  the  wavefronts  formed  in  this  manner 
are  shown  at  Y.  After  traversing  the  lens,  they  will  be  brought 
to  focus  at  a  point  1^  in  the  principal  focal  plane.  This  point 
will  of  course  be  the  intersection  of  the  plane  with  the  un- 
deviated  ray  for  this  set  of  waves,  which  is  a  line  drawn  through 
the  optical  center  0  perpendicular  to  the  wavefronts  Y.  Still 
a  third  set  of  wavefronts  will  be  formed  by  a  combination  of 
the -12th  wavelet  from  a,  the  10th  from  b,  the  8th  from!  c,  etc. 
These,  a  few  of  which  are  drawn  at  Z,  are  still- more  inclined 
to  the  original  wavefronts,  and  are  brought  to  focus  farther 
out  in  the  focal  plane,  at  such  a  point  as  I,. 

It  is  not  difficult  to  prove  that  if  0X  represents  the  angle 
between  the  Y  and  the  X  wavefronts  (which  is  the  same  as  the 
angle  IjOP)  and  62  the  angle  between  the  Z  and  X  wavefronts 
(the  angle  Tv>OP),  A  the  wavelength  of  the  light,  and  <r  the 
distance  between  the  centers  of  slits  in  the  grating, 

A  2A 

sin.  6t  =  -  sin.  02  =  — 

a  a 

Of  course,  the  bright  points  on  one  side  of  P  would  be  dupli- 
cated by  corresponding  bright  points  on  the  other  side,  since 
everything  is  symmetrical  about  the  axis  of  the  figure.  It  is 
also  possible  that  there  may  be  other  bright  points,  for  which 
the  sine  of  the  angle  is  3A/0-,  4A/<r,  etc.  There  is,  however,  a 
limit  to  the  number  of  bright  points  obtainable;  for  the  sine 
of  an!  angle  cannot  be  greater  than  1,  and  if,  for  instance, 
4A<o-<5A,  there  will  be  four  bright  points  on  each  side  of  P, 
but  not  5. 

The  formation  of  the  wavefronts  by  monochromatic  light 
in  passing  through  the  grating  can  be  very  nicely  illustrated  by 
the  following  experiment  with  ripples  in  a  basin  of  mercury. 


124 


LIGHT 


To  one  prong  of  a  tuning-fork  is  fastened  a  piece  of  sheet- 
metal  cut  like  a  comb  with  some  16  teeth  spaced  about  3/16 
inch  apart,  as  shown  in  figure  67.  The  fork  is  then  placed  so 

that  the  teeth  just  dip 
below  the  surface  of  the 
mercury  in  the  basin,  and 
is  set  into  vibration.  Each 
tooth  becomes  a  center  for 
a  series  of  ripples  which 
Figure  67  emanate  from  the  teeth 

just  as  secondary  light  wavelets  emanate  from  the  openings  in 
the  grating  of  figure  66.  The  only  difference  is  that  the  teeth 
are  the  actual  sources  of  the  ripples,  while  the  openings  in  the 


Figure   68 

grating  only  become  centers  of  wavelets  because  the  incident 
plane  wavefronts  bring  the  disturbance  up  to  them.  The  ap- 
pearance of  the  mercury  surface  becomes  like  figure  68,  which 
is  an  instantaneous  photograph.  The  heavy  dark  place  of 


PLANR  GRATING  125 

irregular  shape  is  the  shadow  of  part  of  the  tuning-fork.  Close 
to  the  comb,  the  vibrations  are  too  complicated  to  be  analyzed, 
but  farther  away  five  separate  sets  of  plane  wavefronts  can  be 
clearly  seen.  The  central  one,  X,  advancing  perpendicular  to 
the  comb,  corresponds  to  the  wavefronts  X  of  figure  66.  The 
two  marked  Y  correspond  to  the  Y  wavefronts  of  figure  66  and 
a  symmetrical  set  on  the  other  side  of  the  axis,  and  similarly 
for  those  marked  Z.  If  the  wavelength  of  the  ripples  had  been 
somewhat  less,  or  the  space  between  teeth  greater,  other  sets 
of  wavefronts  might  have  been  seen.  The  comb  was  so  con- 
structed that  2A<<r<3A,  so  that  only  two  spectra  could  be 
expected  on  each  side  of  the  central  axis. 

It  was  not  practicable  in  the  case  of  the  ripples  to  arrange 
any  device  for  focussing,  and  one  may  say  that  figure  68  cor- 
responds to  figure  66  with  the  lens  removed. 

Now,  suppose  that  the  incident  light  contains,  beside  the 
wavelength  already  considered,  A,  another,  longer  or  shorter, 
which  we  shall  call  A'.  The  grating  would  produce  several  sets 
of  wavefronts  for  this  wavelength  also.  The  central  set  would 
be  parallel  to  the  incident  wavefronts  C,  and  would  therefore 
be  brought  by  the  lens  to  the  principal  focus  P,  like  those  for 
wavelength  A.  The  other  sets  for  A',  however,  would  not  be 
brought  to  the  same  points  as  the  corresponding  sets  for  A,  for 
we  have  seen  that  the  angles  QL,  02,  etc.,  depend  upon  the  wave- 
length. If  A'>A,  each  bright  point  for  A'  will  be  focussed 
farther  from  P  than  the  corresponding  point  for  A,  and  con- 
versely if  A'<A.  Therefore,  however  many  wavelengths  may  be 
present  in  the  incident  light,  a  series  of  spectra  will  be  formed 
on  each  side  of  the  central  spot  P.  These  are  known  as  the 
first  spectrum,  second  spectrum,  etc.,  to  the  right  or  left,  as 
the  case  may  be.  The  central  spot  P,  since  it  contains  in  itself 
all  wavelengths  of  the  incident  light,  is  sometimes  spoken  of  as 
the  "spectrum  of  order  zero."  This  manner  of  speaking  is 
consistent  with  the  general  formula  for  the  grating, 


For  the  first  spectrum,  n  =  1,  for  the  second  n  =  2,  and  for 
the  spectrum  of  order  zero  n  =  0,  so  that  0  =  0  for  all  wave- 
lengths. 


126 


LIGHT 


In.  order  to  measure  wavelengths  with  the  grating,  we  must 
first  know  <r,  the  distance  between  the  centers  of  the  slits,  some- 
times known  as  the  "grating-space."  This  can  be  found  by 
placing  the  grating  under  a  high-power  microscope  and  measur- 
ing with  a  micrometer.  'Then  we  must  measure  one  of  the 
angles  Oi  or  #,,  etc.,  for  the  wavelength  desired.  If  the*  lens 
of  figure  66  is  th|e  objective  of  a  telescope,  and  if,  as  in  the 
figure,  the  axis  of  the  telescope  is  perpendicular  to  the  plane 
of  the  grating,  the  angle  may  be  found  by  means  of  a  mi- 
crometer. But  when,  as  is  usually  the  case,  the  grating  is 
mounted  on  the  table  of  a  spectrometer,  so  that  the  telescope 
may  be  swung  about  an  axis  through  the  grating  perpendicular 
to  the  pla.ne  of  the  diagram,  then  it  can  be  turned  so  as  to 
receive  upon  the  cross-hair  at  the  principal  focus  any  wave- 
length in  any  order  of  spectrum,  so  that  the  angle  0  can  be  read 
off  from  the  verniers  attached  to  the  telescope. 

61.  Why  the  lines  are  sharp, — Of  course,  these  measure- 
ments would  still  be  very  inaccurate,  and  lines,  or  maxima,  due 


Figure    69 

to  different  wavelengths  would  seriously  overlap,  unless  these 
maxima  were  quite  sharp,  as  indicated  in  figure  65  (Y).  It 
remains  to  show  why  the  large  number  of  slits  in  the  grating 
produces  the  desired  sharpness. 

Consider  figure  69.  AB  represents  a  grating,  which  for 
the  sake  of  defmiteness  and  simplicity,  we  shall  suppose  has 
9  slits  or  openings,  although  a  practical  grating  usually  has 


SHARPNESS  OP  THE  LINES  127 

many  thousand.  We  suppose  there  is  only  a  single  wavelength 
present  in  the  light.  The  only  wavefront  drawn,  iy,  is  one 
for  the  first  spectrum.  Rays  are  drawn  from  each  slit,  running 
through  the  lens  to  the  point  I,  where  they  are  brought  to 
focus.  The  lens  is  turned  in  such  manner  that  these  rays  are 
parallel  to  the  lens-axis,  so  that  I  is  really  the  principal 
focus.  From  what  we  have  learned  about  lenses,  it  is  evident 
that  there  is  the  same  number  of  wavelengths  in  each  ray, 
measured  from  the  wavefront  iy  to  the  point  I,  and  brightness 
occurs  at  I  because,  measured  from  the  slits  to  this  wavefront, 
each  ray  has  one  more  wavelength  than  the  one  next  below  it 
in  the  figure,  so  that  all  arrive  at  I  in  the  same  phase.  Now 
let  im  represent  a  hypothetical  wavefront,  slightly  inclined  to 
iy,  so  that  the  perpendicular  distance  am  contains  9  wave- 
lengths, whereas  ay  contains  only  8.  Such  a  wavefront  would 
be  brought  to  focus  at  a  point  x,  very  close  to  I  because  im  is 
so  slightly  inclined  to  iy.  We  shall  show  that  practically  no 
light  reaches  the  point  x  from  the  9  slits  because  the  rays  that 
might  come  there  mutually  interfere,  or  annul  one  another. 
The  proof  is  as  follows:  Between  the  wavefront  im  and  x, 
there  will  be  the  same  number  of  wavelengths  in  every  ray, 
otherwise  x  would  not  be  the  proper  focus  for  a  wavefront  in 
the  position  im.  Hence  we  need  consider  only  those  portions 
of  the  nine  rays  that  lie  between  im  and  the  grating.  There 
are  9  wavelengths  in  am,  9  X  %,  or  7  %  in  bn,  9  X  %  =  6  % 
in  co,  9  X  %  =  5  %  in  dp,  9  X  %  =  4  %  in  eq,  9  X  %  = 
3  %  in  fr,  9  X  %  =  2  V4  in  gs,  9  X  %  =  1  Vs  in  ht.  Then 
am  is  just  4  y2  wavelengths  longer  than  eq,  and  therefore  the 
rays  from  a  and  e  will  interfere  and  annul  one  another,  so  far 
as  effect  at  x  is  concerned.  Similarly,  and  for  the  same  reason, 
the  ray  from  b  will  annul  that  from  f,  the  ray  from  c  that 
from  g,  and  the  ray  from  d  that  from  n.  There  remains  only 
the  ray  from  i,  not  annulled  by  any  other  ray,  to  produce 
illumination  at  x.  In  a  later  chapter,  (see  section  91),  it  will 
be  shown  that  in  such  a  case  the  illumination  is  proportional 
to  the  square  of  the  number  of  effective  elements;  therefore, 
since  9  slits  conspire  in  phase  to  produce  illumination  at  I, 
and  only  1  at  x,  the  illumination  at  the  latter  point  will  be 
only  1/81  that  at  the  former. 


128  LIGHT 

Now,  suppose  that,  instead  of  9  slits,  the  grating  contained 
999.  Then  ay  would  contain  998  wavelengths,  and  we  should 
draw  im  so  that  am  contained  999.  The  point  x  would  then 
come  very  close  indeed  to  I,  and  the  brightness  of  illumination 
at  x  would  be  only  l/(999)2  that  at  I,  because  only  one  open- 
ing in  the  grating  would  contribute  to  it,  all  the  others  annul- 
ling one  another's  effects  in  pairs.  We  should  then  be  fully 
justified  in  saying  that  x  is  practically  a  dark  point,  and  it  is 
quite  easy  to  prove,  by  similar  reasoning,  that  there  is  another 
point  x',  close  to  the  other  side  of  I,  which  is  equally  dark. 
Since  there  is  a  dark  spot  so  very  close  on  each  side  of  the 
bright  spot  I,  the  maximum  of  illumination  must  be  very  sharp. 
Moreover  since,  if  the  grating-space  o-  is  quite  small,  the  max- 
ima of  two  successive  orders,  such  as  I:  and  I2  of  figure  66  are 
quite  far  apart,  we  have  just  the  conditions  for  accurate 
measurements,  viz.,  sharp  and  widely  separated  maxima  of 
brightness.*  In  our  proof,  we  have  taken  only  the  case  of  a 
grating  having  an  odd  number  of  openings,  but  the  result 
would  be  the  same  in  all  essentials  if  the  number  were  even. 
In  fact,  the  dark, points  would  then  be  absolutely  dark,  instead 
of  only  relatively  so. 

In  actual  gratings  there  are  sometimes  as  many  as  120000 
openings,  and  the  grating-space  a  has  been  made  as  small  as 
1/20000  inch,  though  1/15000  inch  is  more  common.  It  is  of 
course  obvious  that  actual  slits  could 'not  be  cut  so  close  to- 
gether as  this,  through  a  solid  plate.  Gratings  are  made  by 
ruling  upon  a  plate  of  glass  an  immense  number  of  fine  parallel 
scratches  with  a  diamond.  One  may  regard  the  scratches  as 
being  opaque,  the  clear  unscratched  spaces  between  taking  the 
place  of  the  slits.  Strictly  speaking,  this  is  incorrect,  for  the 
scratched  grooves  are  really  not  opaque.  But  the  diminished 
thickness  of  the  glass  where  the  grooves  are  cut  causes  the 
light  passing  through  these  places  to  be  out  of  phase  with  that 
which  passes  through  the  clear  spaces,  and  this  has  the  same 
effect,  so  far  as  the  location  of  the  spectra  is  concerned,  as  if 
the  grooves  were  really  opaque. 

*A  more  thorough  treatment  shows  that  between  the  sharp  bright 
maxima  of  different  orders  there  are  a  great  many  faint  secondary 
maxima,  but  with  practical  gratings  these  are  tno  feeble  to  be  observed. 


CONCAVE  GRATING  129 

62.  Reflection  gratings. — The  best  gratings  are  ruled,  not 
on  glass,  but  on  a  polished  plate  of  metal.    In  such  cases,  it  is 
the  reflected  light  that  produces  the  spectra.    A  study  of  figure 
66  will  show  that  if  the  spaces  a,  b,  c,  etc.,  reflected  light,  while 
the  spaces  between  did  riot,  spectra  would  be  formed  in  exactly 
the  same  way,  but  on  the  opposite  side  of  the  grating.     The 
great  difficulty  about  a  grating    is    to    make    the    rulings,    or 
''slits,"  perfectly  uniform  in  spacing.     This  involves  a  great 
many  practical  difficulties  which  it  is  impossible  to  overcome 
entirely.     Still,  a  good    grating    gives    beautifully    sharp    and 
clear  spectral  lines,  making  it  much  superior  to  a  prism  for 
producing  spectra,  except  in'  one  particular, — since  a  grating 
produces  many  spectra,  no  one  of  them  contains  more  than  a 
fraction  of  the  available  light,  and  therefore  grating  spectra 
are  in  general  not  so  bright  as  those  produced  by  prisms. 

63.  The  concave  grating. — Henry  Augustus  Rowland,  who 
developed  the  manufacture  and  use  of  gratirigs  for  the  measure- 
ment of  wavelengths    and    other    spectroscopic    investigations, 
conceived  the  idea  of  ruling  gratings  upon  concave  reflecting 
surfaces,  in  order  to  avoid  the  use  of  lenses  to  focus  the  light. 
This  is  of  particular  importance  in  the  regions  of  very  short 
wavelength,  for  glass  absorbs  much  of  the  ultraviolet  light,  and 
it  is  difficult  to  make  satisfactory  lenses -out  of  materials  which 
are  free  from  this  objectionable  quality.     The  idea  proved  very 
valuable,  and  the  so-called  ''concave  grating"  is  a  most  use- 
ful instrument,  simple  and  convenient  in  manipulation.     For 
the  theory  of  the  concave  grating,  the  student  should  consult 
some  more  advanced  text,  such  as  Preston  'a  '  *  Theory  of  Light, ' ' 
or  Baly's  "  Spectroscopy, "  where  the  following  relations  are 
proved : 

Suppose  that,  in  a  plane  through  the  middle  of  the  grating 
and  perpendicular  to  the  rulings,  a  circle  be  drawn  whose 
diameter  is  the  radius  of  curvature  of  the  concave  surface,  so 
that  the  circle  is  tangent  to  that  surface  at  one  end  of  the 
diameter,  and  passes  through  the  center  of  curvature  at  the 
other  end.  Then,  if,  the  slit  be  placed  anywhere  on  this  circle, 
the  different  spectra  will  be  sharply  focussed  at  various  points 
about  the  same  circle.  Moving  the  slit  will  of  course  cause  the 
spectra  to  shift  their  position,  but  so  long  as  the  former 


130 


LIGHT 


remains  on  the  circle,  the  latter  will  also.  Rowland  adopted  a 
very  simple  and  ingenious  method  for  making  sure  that  the 
slit  remains  on  this  focal  circle,  as  shown  by  diagram  in  figure 
70.  Two  beams,  AS  and  BS,  are  set  up  making  an  angle  of 


Figure  70 

90°.  The  slit  is  placed  at  the  intersection  of  these  two  beams, 
i.  e.,  at  S.  The  grating  G,  and  a  plateholder  P,  to  hold  a  long 
narrow  photographic  plate,  are  mounted  at  opposite  ends  of  a 
third  beam,  whose  length  is  equal  to  the  radius  of  curvature 
of  the  grating.  Thus,  the  center  of  curvature  of  the  grating 
comes  just  at  the  middle  of  the  photographic  plate.  The  ends 
of  the  beam  GP  are  mounted  011  slides  or  carriages,  so  that 
the  end  G  can  slide  along  the  beam  BS  while  the  end  P  slides 
along  the  beam  AS.  This  beam  therefore  always  forms  the 
hypothenuse  of  a  right-angled  triangle,  the  acute  angles  of 
which  can  be  changed.  From  the  geometrical  proposition  that 
an  angle  inscribed  in  a  semi-circle  is  a  right-angle,  it  follows 
at  once  that  S  always  remains  on  a  circle  of  which  GP  is  a 
diameter.  Of  course,  the  photographic  plate  P  is  bent  to  fit 
this  circle. 

64.  The  ultraviolet  region.  Fluorescence.  Phosphores- 
cence. Photography. — It  was  stated  in  section  22  that  there  are 
waves  shorter  than  the  violet,  and  others  longer  than  the  red. 
The  former  are  called  ' ' ultraviolet, "  the  latter  ''infrared." 

Although  ultraviolet  light  does  not  itself  produce  vision, 
it  can,  by  aid  of  the  phenomenon  called  "fluorescence,"  give 
rise  to  visible  light  and  thus  make  its  presence  known.  Fluores- 


THE  INFRARED  131 

cence  is  a  property  of  a  great  many  substances,  including  the 
dyes  fluorescein  and  uranin.  It  is  the  power  to  absorb  certain 
wavelengths  and  re-emit  the  energy  in  longer  wavelengths, 
instead  of  turning  it  into  heat.  Fluorescein  and  uranin,  for 
instance,  absorb  ultraviolet  light  and  re-emit  the  energy  'as  yel- 
lowish-green light.  It  is  very  easy  to  show  the  existence)  of 
waves  beyond  the  violet  in  the  spectrum  of  the  sun  or  of  an 
arc-light,  by  holding  in  that  position  a  thin-walled  flask  con- 
taining a  solution  of  one  of  these  dyes,  or  a  glass  plate  which 
has  been  colored  with  one  of  them. 

There  are  some  substances  which  continue  to  give  off  light 
long  after  the  incident  light  has  been  cut  off.  This  phenomenon 
is  called  "phosphorescence/'  because  it  suggests  the  glow 
shown  by  a  piece  of  phosphorus  that  has  been  rubbed,  in  the 
dark.  The  latter  glow,  however,  is  not  true  phosphorescence, 
because  it  is  caused  by  slow  oxidation,  rather  than  by  previous- 
ly absorbed  light. 

The  principal  method  for  studying  the  ultraviolet  region 
is  by  photography,  for  these  waves  are  particularly  effective  on 
a  photographic  plate.  In  order  to  go  very  far  into  the  ultra- 
violet, however,  it  is  necessary  to  avoid  the  use  of  glass  for 
prisms  and  lenses,  since  it  absorbs  very  strongly  all  but  the 
longer  ultraviolet  waves.  Either  quartz  or  fluorite  are  used 
for  this  purpose,  these  substances  being  transparent  for  much 
shorter  waves  than  glass,  but  for  the  shortest  waves  gratings 
and  mirrors  must  be  used.  Even  a  little  air  absorbs  such 
waves,  and  Schumann  and  Lyman,  who  got  wavelengths  as 
short  as  .00001cm.,  were  obliged  to  work  in]  a  vacuum  or  a 
hydrogen  atmosphere.  Very  recently,  Millikan  and  Sawyer 
have  found  waves  as  short  as  .00000272cm.  from  a  spark  in  a 
high  vacuum. 

65.  The  infrared  region. — Photography  has  also  been  ap- 
plied to  the  study  of  the  shorter  infrared  waves,  for  plates  can 
be  specially  prepared  suitable  for  this  purpose.  But  the  usual 
method,  and  the  only  method  suitable  for  the  very  long  waves, 
is  to  detect  them  by  the  heat  produced  when  they  are  absorbed. 
The  infrared  waves  are  sometimes  erroneously  called  il  heat- 
waves," because  the  major  part  of  the  energy  radiated  from 
the  sun  or  any  hot  solid  is  composed  of  waves  longer  than  the 


132  LIGHT 

red,  and  they  are  therefore  responsible  for  most  of  the  heat 
produced  when  the  light  is  absorbed.  But  it  is  conceivable  that 
some  source  of  light  might  radiate  in  such  a  way  that  most  of 
the  energy  was  in  the  ultraviolet,  in  which  case  it  would  be  the 
very  short  waves  that  were  responsible  for  most  of  the  heating. 
In  fact,  there  is  no  such  thing  as  a  heat-wave,  at  least  in  the 
sense  in  which  we  understand  a  wave  when  speaking  of  light, 
sound,  etc.  For  heat  consists  of  entirely  irregular  molecular 
motions,  which  have  nothing  in  common  with  waves  in  the 
ether  except  that  they  possess  energy.  Light  energy  does  not 
become  heat  energy  till  the  light  has  been  absorbed  and  the 
waves  as  such  have  ceased  to  exist. 

.  The  distribution  of  waves  in  the  infrared,  or  for  that  mat- 
ter also  in  the  visible  and  ultraviolet,  might  theoretically  be 
mapped  out  by  moving  a  thermometer  with  a  small  blackened 
bulb  through  the  whole  focal  plane  of  the  spectrum,  and  noting 
the  therometer  reading  at  each  point,  but  a  mercury  or  alcohol 
thermometer  is  far  too  insensitive  for  such  a  purpose.  The 
devices  that  have  been  most  commonly  used  are  the  bolometer 
and  the  thermopile,  but  any  other  very  sensitive  temperature- 
indicator  might  be  used  in  which  the  object  whose  temperature 
is  measured  is  in  the  form  of  a  strip,  narrow  enough  to  cover 
only  a  small  section  of  the  spectrum  at  a  time,  and  blackened 
so  as  to  absorb  and  turn  into  heat  whatever  radiant  energy 
falls  upon  it. 

66.  The  bolometer. — In  the  bolometer,  this  strip  is  a  thin 
and  narrow  piece  of  blackened  platinum,  which  is  connected 
ap  with  three  other  conductors  so  as  to    form    a    Wheatstone 
bridge,  to  which  a  battery  and  a  very  sensitive  galvanometer 
are  also  connected  in  the  usual  way.     The  bridge  is  balanced 
so  that  there  is  no  deflection    of    the    galvanometer    when    no 
radiation  is  falling  upon  the  strip.     If  the  strip  is  placed  any- 
where in  the  spectrum  so  that  energy  falls  upon  jt,  it  is  heated,, 
its  resistance  is  thereby  changed,    the    bridge    becomes    unbal- 
anced, and  a  deflection  of  the  galvanometer  results.    The  mag- 
nitude of  the  deflection  indicates  the  intensity  of  the  radiation 
of  wavelength  corresponding    to    the    point    in    the    spectrum 
where  the  platinum  strip  is  at  the  moment. 

67.  The  thermopile. — The  thermopile  depends  upon  the 
principle  that  in  a  metallic  circuit  consisting  of  two  dissimilar 


THE  DOPPLER  PRINCIPLE 


133 


metals,  an  electric  current  will  flow  if  one  junction  be  at  a  differ- 
ent temperature  from  the  other.  Usually,  at  least  for  such 
purposes  as  are  now  under  discussion,  the  circuit  is  com- 
pounded of  many  pieces  of  two  different  and  suitable  metals, 
for  instance  a  piece  of  antimony,  then  one  of  bismuth,  another 
of  antimony,  another  of  bismuth/  etc. %  till  there  may  be  a  dozen 
or  more  such  junctions  in  the  whole  circuit.  A  sensitive  gal- 
vanometer is  also  included,  to  detect  and  measure  the  current. 
Every  alternate  junction  is  brought  into  a  line,  close  to  one 
another,  and  these  junctions  are  exposed  to  the  radiation  while 
the  others  are  shielded  from  it.  The  exposed  junctions  in  line 
are  of  course  blackened  so  that  they  will  absorb  and  not  reflect 
the  radiation. 

With  the  infrared,  as  with  the  ultraviolet,  it  is  necessary 
to  avoid  the  use  of  glass.  Usually  a  prism  of  rocksalt  is  em- 
ployed, and  concave  mirrors  are  substituted  for  the  lenses  of 
the  ordinary  spectrometer,  to  focus  the  spectrum  and  to  col- 
Hmate  the  light. 

68.  The  Doppler  principle.  Motion  of  the  stars.— An  inter- 
esting application  of  the  spectroscope  is  to  the  determination 
of  the  rate  at  which  stars  are 
approaching   or    receding   from 
the  solar  system.     This  is  based 
upon  a  principle  known  as  the 
"Doppler    effect,"     which    ap- 
plies not  only  to  light,  but  also 
to    sound   or   any)  other   waVe 
phenomenon.     If    a    source    of 
waves    is    approaching    an    ob- 
server, the  length  of  the  waves  Fieure  71 
which  he  receives  is  less,  if  it  is  receding  from  him  greater,  than 
if  there  is  no  motion. 

The  reason  is  easily  explained  by  figure  71.  Suppose  that 
the  source  of  waves  is  moving  in  the  direction  of  the  arrow  P, 
with  a  velocity  v,  and  let  V  be  the  velocity  of  the  waves  them- 
selves. Bear  in  mind  that  V  depends  only  upon  the  properties 
of  the  wave-carrying  medium,  not  at  all  upon  the  velocity  of 
the  body  emitting  the  waves.  If  that  body  sends  out  a  crest 
when  it  is  at  A,  then  that  crest  will  expand  into  a  growing 


134  LIGHT 

circle  with  A  as  center,  despite  the  fact  that  the  source  moves 
away  from  A  in  the  meanwhile.  Let  A,  B,  C,  etc.,  be  positions 
of  the  source  at  instants  differing  by  a  period,  so  that  a  crest 
is  started  from  each  of  these  points.  Then  at  any  later  instant 
the  wavefronts  will  be  circles,  as  shown,  ,but  not  concentric 
circles,  the  center  of  each  being  displaced  toward  the  right  from 
the  center  of  the  preceding  one  by  the  distance  the  source 
moves  during  one  period.  If  A  represents  the  normal  wave- 
length, as  it  would  be  if  the  source  were  at  rest,  the  period  is 
A/Y,  and  the  motion  of  the  source  during  this  time  is  vA/Y. 
Evidently,  the  wavelength  received  by  an  observer  to  the  left, 
in  the  direction  of  X,  will  be  increased  by  this  amount,  while 
that  received  by  an  observer  to  the  right,  in  the  direction  Y, 
will  be  decreased  by  the  same  amount.  That  is,  the  observed 
wavelength,  to  a  stationary  observer  in  the  line  of  motion,  will 
be 

A'  =  A±          =A-  (1) 


the  upper  or  lower  sign  being  used  according  as  the  source  is 
receding  or  approaching. 

It   may   be,   however,   that 

,  *-"-%  *        the     observer     instead    of    the 

source  is  moving.     In    such    a 
Figure  72  Cas6j  fae  analysis  must  be  some- 

what different,  as  will  be  shown  by  use  of  figure  72.  Let  S  be  the 
position  of  the  stationary  source,  and  0  the  initial  position  of 
the  observer.  Suppose  the  latter  is  moving  away  from  the  source, 
toward  the  right,  and  let  0'  be  his  position  one  second  later, 
so  that  00'  —  v.  Lay  off  also  the  distance  OX  equal  to  V. 
Within  this  last  distance  there  are  V/A  waves,  and  the  observer 
would  have  received  all  these  waves  in  one  second  had  he 
remained  stationary  at  0.  But  since  he  has,  during  the  second, 
moved  to  0',  he  actually  receives  a  number  less  than  this,  for 
the  waves  lying  between  0  and  0'  have  failed  to  reach  him. 
The  number  he  actually  receives  per  second  is  therefore 
(Y  —  v)/A,  and  this  is  the  frequency  of  the  waves  as  he 
receives  them;  whereas  the  frequency  with  which  they  are 
emitted  from  the  source  is  Y/A.  This  case  differs  from  the 
preceding  in  that  the  wavelength  is  not  actually  changed,  but 


THE  DOPPLER  PRINCIPLE  135 

it  seems  to  the  observer  to  be  changed;  for  since  his  spectro- 
scope or  other  device  for  measuring  wavelengths  is  moving 
along  with  him,  it  acts  as  if  the  waves  had  the  velocity  V  with 
respect  to  the  observer  together  with  a  frequency  (V  —  v)/A. 
Since  wavelength  equals  volocity  divided  by  frequency,  the 
spectroscope  therefore  indicates  the  wavelength  to  be 


A        '  V  — v 

If  the  observer  is  approaching  the  source,  instead  of  receding 
from  it,  it  is  easy  to  prove  in  a  precisely  analogous  way,  that 
the  wavelength  appears  to  be  VA/(V  +  v),  or  in  general  the 
apparent  wavelength  is 


the  lower  sign  being  taken  if  the  observer  is  receding,  the  upper 
if  he  is  approaching. 

The  two  formulas,  (1)  and  (2),  are  not  quite  alike,  and 
the  student  is  usually  inclined  to  think  that  they  should  be: 
for  he  argues  that  motion  is  only  relative  and  it  should  make 
no  difference  which  of  the  two  things,  source  or  observer,  is 
at  rest  and  which  in  motion.  But  there  is  a  third  thing  to 
take  into  account,  namely  the  medium  that  carries  the  waves. 
It  must  make  a  difference,  for  instance,  in  the  case  of  sound, 
whether  an  observer  is  moving  through  stationary  air  toward 
a  stationary  sounding  body,  or  the  sounding  body  is  moving 
through  stationary  air  toward  a  stationary  observer.  Just  so, 
it  must  make  a  difference  in  the  analogous  case  of  light,  pro- 
vided that  we  may  regard  the  ether  as  a  material  medium 
which  has  a  definite  location  in  space.  Certain  considerations 
bearing  upon  this  proviso  will  be  brought  out  later  (section 
85). 

For  all  applications  to  spectra,  formulas  (1)  and  (2), 
though  not  identical,  yield  results  so  nearly  identical  that  there 
is  no  distinguishing  between  them.  If  we  divide  the  numerator 
by  the  denominator  in  the  second  member  of  (2)  we  get 


136  LIGHT  - 

which  differs  from  (1)  only  by  the  terms  in  the  square  and 
higher  powers  of  v/V.  The  greatest  velocity  found  in  a  star 
is  of  the  order  of  300  kilometers  per  second,  that  of  light  is 
300000  kilometers;  per  second,  and  therefore  v/V  is  of  the 
order  1/1000.  Since  this  makes  v2/V2  of  the  order  1/1000000 
and  the  other  terms  still  smaller,  they  are  beyond  the  limits  of 
measurement  and  can  just  as  well  be  dropped  out. 

If  a  star  then  is  moving  toward  us,  or  we  toward  it,  all 
the  wavelengths  in  its  light  are  apparently  shortened,  that  is 
all  the  lines  in  its  spectrum  are  shifted  toward  the  violet  end, 
and  if  the  star  is  moving  away  from  us,  or  we  from  it,  the 
lines  are  shifted  toward  the  red.  In  either  case,  the  amount 
of  the  shift  is  proportional  to  the  relative  velocity  or  rather 
to  that  component  of  it  which  is  along  the  line  joining  star  and 
observer.  It  is  possible,  from  the  great  array  of  lines  shown 
in  the  spectra  of  many  stars  to  identify  them  with  certain 
elements  in  spite  of  their  displaced  position,  and  thus  to  know 
their  proper  wavelengths.  To  determine  the  velocity  then,  it 
is  only  necessary  to  photograph  the  star's  spectrum  side  by 
side  with  the  spectrum  of  some  terrestrial  source  such  as  that 
of  a  hydrogen  vacuum  tube  or  a  spark  between  iron  terminals, 
and  measure  the  apparent  wavelengths  by  comparison.  Ob- 
viously, one  cannot  say  whether  a  certain  measured  shift  of  the 
lines  is  to  be  attributed  to  motion  of  the  star  or  of  the  earth,, 
or  partly  to  each,  except  that  we  know  the  earth  to  be  moving 
with  a  tolerably  great  velocity  in  its  orbit  about  the  sun.  The 
sun  itself  may  also  be  moving,  but  it  is  customary  to  treat  all 
such  measurements  of  line-of -sight  velocities  as  if  the  sun  were 
at  rest.  The  measured  velocity  is  corrected  for  the  earth's 
orbital  motion,  and  the  result  stated  as  the  velocity  of  the  star 
"with  respect  to  the  sun." 

It  .is  senseless  to  ask  if  the  sun  has  any  "absolute"  motion, 
for  the  motion  of  one  body  cannot  be  intelligently  thought  of 
except  with  reference  to  some  other  body,  but  one  cannot  help 
speculating  asi  to  whether  the  sun  is  moving  or  at  rest  with 
respect  to  the  ether,  though  such  a  question  presupposes  that 
the  ether  is  to  be  regarded  as  having  a  definite  location  in 
space.  In  section  84  a  certain  experiment  will  be  discussed 
which  bears  upon  this  question. 


THE  DOPPLER  PRINCIPLE  137 

Problems. 

1.  Suppose   the   slit  of   a   spectroscope   to  be   5mm.  long, 
the    focal    length    of    the    collimator  to  be  30cm.,  and  that  of 
the  telescope  to  be  40cm.    How  wide  will  the  spectrum,  in  the 
focal  plane  of  the  telescope,  be? 

2.  A  dense  flint  prism,  of  angle  60°,  has  the  following 
indices  of  refraction:    1.7774  for  wavelength  .00004713,  1.7695 
for  A  .00005016,  1.7537  for  A  .00005896,  1.7444  for  X  .00006708. 
Plot  a  curve  with  A  as  abscissa,  index  as  ordinate.     How  long 
would  be  the  stretch  of  spectrum,  between  the  longest  and  the 
shortest  of  the  above  wavelengths,  if  the  focal  length  of  the 
telescope  were  30cm.?     (It  will  be  sufficient,  though  not  strict- 
ly correct,  to  consider  that  all  the    light    passes    through  the 
prism  at  minimum  deviation,  and  use  the  formula  of  paragraph 
26). 

3.  Calculate  the  wavelengths  of  the  first  four  lines  of  the 
Balmer  formula  for  hydrogen,  and  compare  with  the  experi- 
mentally found  values  given  in  the  foot-note  to  paragraph  54. 

4.  What  would  be  the  spectrum  of  light  reflected  from  a 
white  wall,  in  daylight?     Under  illumination  by  tungsten-fila- 
ment lamps'? 

5.  Some  of  the  dark  lines  in  the  solar  spectrum  are  due 
to  absorption  in  the  earth's  atmosphere.     How  can  these  be 
distinguished  from  the  true  solar  lines? 

6.  From  the  curves  of  figure  64,  show  why  a  heated  solid 
first  becomes  dull  red,  only  becoming  "white-hot"  at  a  very 
high  temperature.     Also  show    why    incandescent    lamps    are 
more  efficient  when  operated  at  high  temperatures. 

7.  Calculate   the   energy  in   a   "quantum,"   for   light   of 
wavelength  .00003  cm.?  and  of  wavelength  .00008  cm. 

8.  "What  wavelengths  in  the  first,  'second,  and  fourth  order 
spectra  come  at  the  same  place  as  A  .00005000  cm.  in  the  third 
order,  with  a  grating. 

9.  Show  that,  in  a  plane  grating,    if   the   incident   plane 
waves  strike  with  an  angle  of  incidence  i  instead  of  zero,  the 
formula  becomes  sin.  i  ±  sin.  Q  =  nA/<r. 

10.  If  a  certain  star  is  moving  away  from  the  earth  at  a 
speed  of  100  kilometers  per  second,  find  the  change    in    wave- 
length of  a  line  whose  proper  wavelength  is  .000065  cm.  (red). 
Also  for  one  whose  proper  wavelength  is  .000040  cm.  (violet). 


CHAPTER  VIII. 

69.  The  approximately  rectilinear  propagation  of  light. — 70.  Shadow 
of  a  straight  edge. — 71.  Shadow  of  a  wire. — 72.  Diffraction  through  a 
rectangular  opening. — 73.  Resolving-power. 

69.  The  approximately  rectilinear  propagation  of  light, 

We  have  already  seen,  in  section  18,  that  light  does  not  travel 
absolutely  in  straight  lines;  that  is,  it  does  not  cast  absolutely 
sharp  shadows,  even  when  the  light-source  is  as  narrow  as  we 
can  make  it.  In  fact,  we  should  rather  expect  that  light,  like 
other  forms  of  wave  motion,  would  bend  rather  freely  about  an 
obstacle  in  its  path,  and  our  first  interest  lies  in  explaining, 
not  why  it  bends  at  all,  but  why  it  does  not  bend  more. 

The  statement  that  light  travels  approximately  in  straight 
lines  really  amounts  to  this,  that  a  comparatively  small  object, 


Figure  73 

even  when  held  some  distance  from  the  eye,  will  screen  off  all, 
or  at  least  most,  of  the  light  coming  from  a  point.  For  instance, 
a  ten-cent  piece  six  feet  from  the  eye,  will  pretty  effectually 
blot  out  the  light  of  a  star,  although  the  same  coin  at  one  hun- 
dred feet  will  not.  In  this  respect,  light  forms  a  marked  con- 
trast with  sound,  for  even  an  obstacle  several  feet  in  diameter, 

(138) 


HUYGHENS'  ZONES  134 

when  six  feet  away,  produces  very  little  effect  on  the  intensity 
of  a  sound.  Our  first  problem,  then,  will  be  to  explain  why 
the  very  much  shorter  wavelengths  of  light  cause  this  great 
difference.  For  simplicity,  we  shall  assume  that  we  are  to  deal 
with  only  plane  waves,  and  only  monochromatic  light,  that  is. 
light  all  of  one  wavelength.  The  area  ABCD  figure  73  repre- 
sents in  perspective  a  portion  of  a  plane  wavefront  advancing 
from  the  left  toward  the  point  0.  According  to  Huyghen's 
principle,  the  effect  at  0  may  be  regarded  as  made  up  of  the 
summation  of  all  effects  produced  by  secondary  wavelets,  one 
coming  from  each,  point  in  the  wavefront.  Some  of  these 
secondary  wavelets  will  annul  one  another  upon  reaching  O, 
but  there  will  be  a  net  residual  effect,  which  we  shall  prove  is 
the  same  as  if  all  but  a  small  portion  of  the  wavefront  were 
annihilated.  Let  P  be  the  foot  of  a  perpendicular  drawn  from 
0  to  the  wavefront,  and  r  the  distance  OP.  Imagine  a  num- 
ber of  spheres  drawn  with  0  as  center,  and  with  radii  equal 

to      r  +i,  r  4-  x,   r  +  ^,  i •  +  2A,  r  +  5A,  etc.,  increasing  by 

A/2  each  time.  Each  sphere  will  intersect  the  wavefront  in  a 
circle  with  P  as  center,  and  the  whole  wavefront  will  be  divided 
up  into  a  large  number  of  regions  known  as  "Huyghens' 
zones,"  the  innermost  one  a  circle,  the  others  rings.  We  shall 
number  them  1,  2,  3,  etc.,  beginning  at  the  center.  The  most 
important  characteristic  of  these  zones  is  this,  that  the  aver- 
age distance  from  0  to  the  points  in  any  one  zone  is  half  a 
wavelength  shorter  than  that  for  the  next  zone  beyond.  If  we 
consider  the  effect  of  any  one  zone,  at  the  point  0,  it  is  evident 
that  points  on  the  inner  boundary,  being  one-half  wavelength 
nearer  than  those  on  the  outer  boundary,  will  tend  to  neutral- 
ize the  latter 's  effects;  but  points  lying  nearer  the  middle  line 
of  the  ring  will  not  completely  neutralize  one  another,  so  that 
there  will  be  a  certain  net  effect  at  0  due  to  the  whole  zone. 
Each  zone,  however,  tends  to  neutralize  the  effect  of  the  next 
zone  within  it.  Therefore  we  can  represent  the  effects  of  the 
odd  zones  by  positive  quantities,  d,,  ds,  d5,  etc.,  and  those  of 
the  even  ones  by  negative  quantities,  — d2,  — d4,  etc.,  and  the 
effect  of  the  whole  system  of  zones,  that  is,  the  whole  wave- 
front,  will  be  properly  represented  by  a  series, 


140  LIGHT 

D  =  dt  — •  d2  +  ds  —  d4  +  dB  —  d,  4-  etc. 

which  will  have  an  infinite  number  of  terms  if  the  wavefront 
is  not  limited  in  extent. 

We  shall  now  prove  that  dt  is  slightly  greater  than  d2,  d2 
than  d3,  etc.,  so  that  the  series  consists  of  alternate  positive 
and  negative  terms  of  decreasing  magnitude,  and  is  therefore 
convergent  and  has  a  definite  value. 

The  effect  of  any  zone  depends  upon  three  things,  its  area, 
its  distance  from  0,  and  the  inclination  at  which  its  light  comes 
to  0.  Leaving  aside  for  a  moment  the  influence  of  the  inclina- 
tion, the  effect  should  be  proportional  to  the  area  of  the  zone 
divided  by  its  distance  from  0.  If  R  represents  the  radius 
of  the  outer  boundary  of  the  first  zone,  we  have  from  simple 
geometry  R2  +  r2  =  (r  +  V'2)2,  giving 

R2  =  rA  +  A2/4 

therefore  the  area  of  the  first  zone  is  TrR2  or  ir(rX  +  A2/4). 
It  is  easy  to  show  in  a  similar  way  that  the  square  of  the 
radius  of  the  outer  boundary  of  the  second  zone  is  2rA  -j-  A2, 
and  the  area  of  the  corresponding  circle  7r(2rA  +  ^2)-  The 
area  of  the  second  zone  is  the  difference  between  the  areas  of 
these  circles,  or  7r(rA  -j-  3A2/4).  Similarly,  the  area  of  the 
third  zone  is  7r(rA  +  5A2/4),  that  of  the  fourth  7r(rA  +  7A2/4), 
etc. 

The  average  distance  of  points  in  the  first  zone  from  0  is 
r  4-  A/4,  that  of  points  in  the  second  zone  r  +  3A/4,  etc.,  so 
that  if  we  divide  the  area  of  each  zone  by  its  average  distance 
from  O  we  get  in  each  case  the  quotient  ?rA.  Therefore,  the 
effect  of  zone  distance  and  zone  area,  in  causing  a  variation  in 
the  terms  of  the  series,  mutually  counterbalance  one  another, 
so  that  the  only  thing  to  consider  is  the  inclination.  Since  the 
light  from  the  outer  zones  comes  to  the  point  0  at  a  greater 
inclination  with  the  line  PO  than  does  that  from  the  inner 
zones,  it  follows  that  the  magnitude  of  the  terms  d1?  d2,  etc., 
continuously  decreases  with  higher  order,  and  the  series  is 
convergent.  The  difference  between  successive  terms,  however, 
is  quite  small,  particularly  in  the  case  of  the  first  few  terms. 

The  easiest  way  to  get  an  idea  of  the  value  of  the  whole 
series,  representing  the  effect  of  the  whole  wavefront  upon  the 


HUYGHENS'  ZONES  141 

point  0,  is  by  a  graphical  method.  Lay  off  on  a  straight  line, 
as  in  figure  74.  the  distance  oa  equal  to  dt.  From  a  lay  off  in 
the  opposite  direction  the  distance  • 

ab,     equal     to     d,.      Then    ob  =       J_+-f ]  * 

dt— d2.     Lay  off  be  equal  to  d31 
cd  equal  to  d4,  and  so  on.     Then  Figure  74 

oc  =  d,  —  d2  +  d3,  od  =  d,  —  d2  +  d3  -  -  d4,  and  so  on.  It  is 
evident,  that,  since  each  term  is  smaller  than  the  one  pre- 
ceding it,  the  whole  series  will  have  a  value  less  than  d,, 
and  if  we  make  an  assumption  about  the  manner  in  which  the 
terms  decrease  which  is  certainly  reasonable,  it  can  be  shown 
that  the  sum  of  an  infinite  number  of  terms  will  be  just  equal 
to  half  of  d,.  The  physical  meaning  of  this  mathematical  result 
is  that  the  effect  of  vtbe  whole  wavefront  is  only  half  that  which 
the  first  zone  alone  would  exert,  so  that  if  an  opaque  screen 
were  placed  in  the  path  of  the  light,  with  a  hole  in  it  only 
large  enough  to  let  the  first  zone  through,  the  illumination  at 
O  would  be  actually  increased.  This  surprising  prediction  is 
actually  verified  by  experiment,  but  it  must  be  remembered  that 
it  is  only  at  the  point  0,  and  in  its  immediate  neighborhood, 
that  the  illumination  is  increased.  Other  points  in  the  plane 
of  0  become  darker.  Furthermore,  since  the  size  of  a  zone 
depends  not  only  upon  the  wavelength,  but  also  upon  the  dis- 
tance r,  a  hole  big  enough  toi  let  through  only  the  first  zone 
as  calculated  for  the  point  0  would  let  through  the  first  and 
second  as  calculated  for  some  point  on  OP  nearer  to  P,  the 
first,  second  and  third  for  a  point  still  nearer,  etc.  Therefore, 
at  a  certain  point  the  illumination  would  be  dt  —  d2,  or  nearly 
zero,  while  at  another  it  would  be  dt  —  c^  +  d3,  nearly  equal 
di  or  d,,,  and  so  on.  Therefore,  different  points  along  the  line 
OP  should  be  alternately  bright  and  dark.  This  statement  also 
is  found  to  be  true. 

The  problem  with  which  we  began  was  to  explain  why  a 
comparatively  small  obstacle  will  cut  off  light  but  not  sound. 
Suppose  then  that  instead  of  an  opaque  screen  with  a  small 
hole,  we  place  in  the  path  of  the  light  a  small  opaque  disc, 
which  will  cut  out  some  of  the  central  zones  but  allow  all  the 
rest  to  pass  on.  For  numerical  calculation,  take  the  diameter 
of  the  disc  to  be  1  cm.,  and  the  distance  from  the  point  whose 


142  LIGHT 

brightness  we  are  considering  to  be?  200  cm.  Let  n  be  the 
number  of  zones  cut  out.  The  square  of  the  radius  of  the  nth 
circle  is  nrA  +  n2A2/4,  and  this  must  be  equal  to  the  square 
of  .5  while  r  is  200  and  A  may  be  taken  as  .00005,  about  the 
brightest  part  of  the  spectrum.  Then, 

(.5)2  =  .25  =  .Oln  +  .000000000625n2 

The  coefficient  of  n2  is  so  small  that  we  may  neglect  it,  since 
we  want  only  the  approximate  value  of  n,  which  is 

n  =  .257.01  =  25 

Since  the  first  25  zones  are  eliminated,  the  illumination  at  0 
will  be  given  by  the  series 

D  =  d20  —  d27  +  d28  —  d29  +  etc. 

whose  value  is  approximately  equal  to  half  the  first  term,  that 
is  doG/2.  Now  although  the  values  of  the  d's  decrease  rather 
slowly,  still  the  26th  term  is  very  much  smaller  than  the  first, 
consequently  the  illumination  at  0  is  very  faint,  though  not 
absolutely  zero. 

As  a  comparison,  let  us  calculate  how  large  a  disc  would 
be  required  to  screen  off  25  zones  from  a  sound  wavefront,  200 
cm.  away,  the  wavelength  being  taken  as  64.45  cm.,  which  cor- 
responds to  a  sound  of  512  vibrations  per  second.  Letting  x 
represent  the  required  radius  of  the  disc,  and  using  the  formula 

x2  =nrA  +  n2A2/4 

we  get  x  =  985  cm.,  that  is,  the  disc  must  be  nearly  20  meters 
in  diameter.  The  example  shows  how  the  great  difference  in 
wavelengths  between  light  and  sound  accounts  for  the  differ- 
ence in  the  effectiveness  of  a  small  obstacle. 

70.  Shadow  of  a  straight  edge. — We  are  now  in  a  position 
to  explain  the  bright  and)  dark  bands  that  appear  near  the 
shadow  of  an  obstacle  of  considerable  width  with  a  sharp 
straight  edge,  which  were  mentioned  in  section  18.  Figure  75 
is  the  same  as  figure  16,  with  the  addition  of  the  wavy  line  from 
X  to  Z,  which  is  a  graph  representing  the  distribution  of 
illumination  as  it  appears  on  the  screen  AB.  The  peak  p  indi- 
cates that  at  the  point  P  on  the  screen  the  illumination  is 


SHADOW  OF  A  STRAIGHT  EDGE 


143 


particularly  bright,  that  is,  there  is  a  bright  band  running 
along  parallel  to  the  edge  e,  at  a  distance  PE  from  what  would 
be  the  edge  of  the  shadow  if  light  did  travel  absolutely  in 
straight  lines.  On  the  other  hand,  the  drop  in  the  curve  to  the 
right  of  C  indicates  that  C  is  relatively  dark,  and  a  dark  band 


Figure   75 

runs  along  the  screen  there,  also  parallel  to  the  edge.  The 
other  peaks  and  depressions  in  the  curve  show  that  there  are 
a  series  of  bright  and  dark  bands,  becoming  less  and  less  pro- 
nounced at  greater  distances  from  the  edge,  until,  at  some  point 
beyond  X,  all  traces  of  bands  are  lost,  and  the  screen  appears 
uniformly  illuminated,  just  as  if  the  obstacle  at  e  were  removed. 

Consider  first  the  illumination  at  the  point  E,  just  on  the 
straight  line  through  the  slit  S  and  the  edge  e.  If  we  take  the 
plane  of  the  wavefront  just  as  it  reaches  the  edge  e,  and  con- 
struct on  it  the  Huyghens'  zones  for  the  point  E,  the  center 
of  the  zones  will  lie  at  e,  and  half  of  each  zone  will  be  cut  off 
by  the  obstacle.  Therefore,  the  effect  at  E,  instead  of  being 
half  that  due  to  the  first  complete  zone,  will  be  only  %  that  of 
the  first  complete  zone,  that  is,  E  will  be  distinctly  darker  than 
if  the  obstacle  were  not  present. 

For  a  point  further  up  on  the  screen,  such  as  P,  the  cen- 
ter of  the  zone  system  would  be  at  such  a  place  as  b.  Let  us 
suppose  that  P,  and  therefore  b,  are  high  enough  so  that  the 
whole  of  the  first  zone  is  uncovered,  but  that  the  obstacle  cuts 
off  a  segment  of  each  of  the  others.  Then  the  effect  at  P  is 
greater  than  it  would  be  with  the  obstacle  removed.  For  we 
have  seen  that  the  effect  of  all  the  zones  from  the  second  on, 


144  LIGHT 

• 

when  they  are  complete,  is  to  nullify  half  the  effect  of  the  first 
zone.  In  this  case  the  first  zone  is  complete  and  the  others 
incomplete,  and  therefore  these  are  incapable  of  nullifying  half 
the  effect  of  the  first.  Thus  the  first  bright  band  at  P  is  ex- 
plained. 

At  a  still  higher  point,  as  C,  the  first  two  zones  are 
clear,  the  rest  partly  covered.  Here  the  second  zone,  being  less, 
opposed  than  it  would  normally  be  by  the  zones  of  higher 
order,  is  more  free  to  oppose  the  action  of  the  first,  therefore 
the  illumination  at  C  is  less  than  if  the  obstacle  were  not  in 
its  place,  and  the  first  dark  band  is  explained. 

The  second  bright  band  occurs  where  three  zones  are  com- 
pletely free,  the  second  dark  one  where  four  are  free,  etc. 
Evidently  these  bands  become  weaker  and  weaker  as  more  and 
more  zones  are  uncovered,  since  the  zones  of  higher  order  are 
weaker  than  those  of  lower. 

Now  take  a  point  D,  below  B.  For  this  point,  the  center 
of  the  zone  system  is  hidden  by  the  obstacle,  and  no  zone  would 
be  complete.  In  this  case  it  is  better  to  construct  the  zones 
on  an  entirely  new  plan.  Instead  of  taking  a  line  from  D 
perpendicular  to  the  wavefront  as  the  basic  distance  for  draw- 
ing the  spherical  surfaces  that  cut  out  the  zones,  we  take  De, 
the  shortest  distance  to  that  part  of  the  wavefront  -that  is  not 
hidden,  and  draw  our  spheres  of  radius  De  +  A/2,  De  +  ^ 
De  -f-  3A/2,  etc.  All  of  the  resulting  zones  will  be  incomplete 
segments  of  rings,  and  the  resulting  illumination  at  D  will  of 
course  be  faint.  The  farther  dowrn  D  is  moved  into  the  shadow, 
the  fainter  it  will  be,  but  there  will  be  no  maxima  or  minima 
of  illumination.  The  brightness  fades  out  continuously  and 
rather  rapidly,  becoming  inappreciable  at  some  such  point  as 
Z.  From  there  on,  no  illumination  can  be  seen  in  the  shadow. 

The  formation  of  bands  at  the  edge  of  a  shadow,  together 
with,  a  number  of  similar  phenomena,  are  classed  under  the 
general  name  of  diffraction.  Bands  like  those  of  figure  75  are 
always  formed  about  the  edges  of  shadows,  whenever  the 
source  of  light  is  small  enough,  or  when,  if  not  small,  it  is 
far  enough  distant  to  subtend  a  very  small  angle,  as  in  the 
ease  of  a  street  light  fifty  or  one  hundred  yards  from  the 
opaque  obstacle  forming  the  shadow.  The  source  may  be  a  long 


DIFFRACTION  THROUGH  AN  OPENING 


145 


line  of  light,  or  a  slit,  if  it  is  parallel  to  the  edge  of  the  obstacle, 
in  which  case  of  course  the  bands  will  be  brighter. 

71.  Shadow  of  a  wire. — One  of  the  most  interesting  cases 
of  diffraction  occurs  when  the  obstacle    is    a    narrow    straight 
object,  such  as  a  wire.    If  a  slit  is  used  as  source,  it  must  be 
parallel  to  the  wire,  and  the  shadow  may  be  cast  on  a  whitened 
wall  or  a  sheet  of  paper.    Bands  are  seen  on  the  outside  of  the 
shadow,  like  those  of  figure  75,  and  in  addition  there  are  always 
one  or  more  bright  bands  running  through  the  center  of  the 
shadow,  formed  by  interference  between  the  light  bending  into 
the  shadow  from  the  two  sides. 

72.  Diffraction  through  a  rectangular  opening. — A  very 
important  case  of  diffraction,  having    much    to    do    with    the 
effectiveness  of  telescopes,  microscopes,  spectroscopes,  and  other 
optical  instruments,  is  the  kind    produced    when    light    passes 
through  a  limited  opening.     We  shall  take  up  only  a  case  in 
which  the  conditions  are  somewhat  simplified,  so  as  to  make 
the  theoretical  discussion  easy. 


] 

c 

L 

1 

1 

-Ul* 

P 

is=_-_-.  ......         -y 

.... 


Figure   76 


In  figure  76,  X  represents  a  system  of  plane  waves  ad- 
vancing toward  the  lens  LL.  The  waves  may  come  from  a  star 
or  a  distant  point  of  light,  or  they  may  come  from  a  slit  placed 
at  the  principal  focus  of  another  lens,  not  shown  in  the  diagram. 
The  lens  LL  tends  to  concentrate  the  light  at  its  principal 
focus  P,  on  the  screen  AB.  cd  is  an  opening,  like  a  moderately 
wide  slit,  in  an  opaque  screen,  so  that  the  only  light  that 
reaches  the  lens  is  in  a  beam,  as  wide  as  cd  in  the  plane  of  the 
paper,  and  as  long  as  may  be  desired  in  a  direction  at  right 
angles  to  the  paper.  That  is,  the  opening  is  in  the  form  of  a 
slit  with  straight  sides,  whose  width  cd  is  not  great.  It  is 
found  that  the  light  is  not  all  brought  to  the  point  P,  but — if 
the  source  is  also  a  slit — there  is  a  bright  band  at  P,  parallel 


146  LIGHT 

to  the  slit-source,  flanked  on  each  side  by  a  number  of  fainter 
bands.  If  the  aperture  cd  is  widened,  the  central  bright  band 
becomes  brighter  and  narrower,  and  the  fainter  bands  become 
much  more  closely  spaced.  If,  on  the  other  hand,  the  aperture 
is  narrowed,  the  central  bright  band  at  P  becomes  widened  and 
the  whole  system  of  bands  widens  out  and  becomes  fainter.  The" 
explanation  is  as  follows: 

Since  the  principal  focus  P  is  not  the  only  point  that 
receives  light  from  the  section  of  wavefront  cd,  we  shall  select 
a  point  at  random,  P',  in  the  focal  plane,  and'  find  what  illumi- 
nation comes  to  it.  Let  0  represent  the  angle  between  the  axis 
of  the  lens  and  the  line  from  the  optical  center  to  P'.  P'  will 
be  a  point  of  absolute  darkness  only  if  all  the  secondary  wave- 
lets, coming  from  every  point  in  the  section  of  wavefront  cd, 
neutralize  one  another.  Draw  de  perpendicular  to  QP'  and  ce 
parallel  to  it.  Let  f  be  the  center  of  the  opening  cd.  From 
our  previous  study  of  lenses,  we  know  that  a  wavefront  de 
would  be  focussed  at  P',  showing  that  from  the  line  de  on  to 
P'  there  are  the  same  number  of  wavelengths  in  every  ray. 
Then  whatever  differences  in  path  exist  in  the  secondary  wave- 
lets are  accounted  for  in  the  space  between  cd  and  de.  If  ce 
is  just  a  wavelength,  the  point  P'  will  be  dark,  exactly  contrary 
to  what  one  would  at  first  thought  suspect.  For  we  must  re- 
member that  not  only  the  points  c  and  d.  but  every  point 
between  c  and  d  sends  a  ray  toward  P7.  For  each  point  in  the 
half  df,  there  is  a  corresponding  point  in  the  half  fc  whose  ray 
has  a  half -wavelength  farther  to  travel  in  getting  to  P'.  There- 
fore one  half  of  the  wavefront  cd  neutralizes  the  effect  of  the 
other  half,  and  P'  is  dark.  If  ce  is  two  wavelengths,  P'  will 
again  be  dark,  for  cd  may  then  be  divided  up  into  four  equal 
parts.  The  upper  two  of  these  will  annul  one  another,  each 
point  in  one  being  A/2  farther  from  P'  than  a  corresponding 
point  in  the  other,  and  for  the  same  reason  the  lower  two  will 
annul  oiie  another.  Similarly,  if  ce  is  equal  to  any  integral 
number  of  wavelengths,  except  zero,  the  point  P'  will  be  dark. 
If  ce  =-  0,  P'  will  of  course  coincide  with  P,  all  the  secondary 
wavelets  will  have  the  same  distance  to  travel,  and  the  point 
will  be  the  brightest  possible.  On  the  other  hand,  if  ce  is 
equal  to  an  odd  number  of  half  wavelengths,  complete  darkness 
at  P'  is  impossible.  Suppose,  for  instance,  that  it  is  3A/2. 


DIFFRACTION  THROUGH  AN  OPENING          147 

Then  cd  could  be  divided  into  thirds.  Each  point  in  the  upper- 
most third  would  annul  the  effect  of  a  point  in  the  middle 
third,  being  one-half  wavelength  farther  from  P',  and  thus  the 
two  upper  thirds  would  cancel  one  another.  The  lowermost 
third,  however,  would  have  a  residual  effect  at  P'  and  would 
therefore  cause  a  small  illumination  there,  which  accounts  for 
the  first  maximum  of  brightness  on  the  lower  side  of  the  prin- 
cipal focus.  Another  maximum  would  occur  at  such  a  position 
of  P'  that  ce  =  5A/2,  etc.  Since  the  angle  ode  =  0,  and  ce 
=  cd.  sin.  0,  we  may  express  what  we  have  found  in  the  form 
of  equations  as  follows:  Complete  darkness  occurs  at  such 
angles  0  that 

sin.  o  =  nA/cd 

where  n  is  any  integer  except  0.  The  greatest  brightness 
occurs  when 

sin.  0  =  0 

but  secondary  maxima  of  brightness  occur  when 
sin.  0  =  nA/2cd 

where  n  is  any  odd  integer  except  1.      (When  ce  =  A/2  the 

point   is  not   dark,   but  neither   is   it   a  3A 

point    of    maximum  brightness.)     From 

the  symmetry  of  the  figure,  it  is  plain 

that  the  points  of  maximum  brightness 

or  of  darkness  on  the  lower  side  of  the 

principal  focus  would  be  duplicated  by 

similar  points,  symmetrically  placed,  on 

the  upper  side.     The  graph  of  figure  77 

shows  the  distribution  of  brightness  on 

Figure  77 

the  screen.    The  symbols  to  the  right  of 

the  vertical  line  indicate  the  values  of  ce  at  the  corresponding 
points  on  the  screen,  while  to  the  left  are  plotted  the  corre- 
sponding brightness. 

If  the  screen  is  removed,  and  an  eyepiece  placed  in  posi- 
tion to  focus  on  the  focal  plane  of  the  lens  LL,  so  that  the 
latter  with  the  eyepiece  form  a  telescope,  then  the  bright  and 
dark  bands  are  seen  in  the  eyepiece  just  as  they  would  be  on 
the  screen,  with  whatever  magnification  the  eyepiece  produces. 


2X 

-3X/2 
X 


-3X/2 
2X 

-5A/2 
3X 


148  LIGHT 

If  the  aperture  cd  is  quite  wide,  the  bands  are  very  fine  and 
sharp,  so  that  a  high  magnification  is  necessary  to  see  them. 

73.    Resolving-power—  The  importance  of  the  considera- 
tions in  the  preceding  section  can  be  understood  from  the  fol- 
lowing example.    Suppose  that  in  figure  76,  instead  of  a  single 
set  of  plane  wavefronts  X,  we  have  two  sets,  nearly  but  not 
quite  parallel  to  one  another,  say  with  a  small  angle  a  between 
them.    For  instance,  the  light  might  be  sodium  light  which  has 
passed  through  a  prism  or  grating,  so  that  the  two  distinct 
wavelengths  present  in  this  light  will  have  their  wavefronts 
slightly  inclined  to  one  another,  the  rays  of  course  also  being 
inclined  to  one  another  at  the  same  angle.     Even  if  there  is 
no  diaphragm  such  as  cd  to  limit  the    two    beams,    still    the 
limited  size  of  the  prism  itself  puts  a  limit  on  the  width  of  the 
beams,  and  the  result  will  be  the  same  as  if  there  were  an 
opening  just  large  enough  to  pass  them.     Therefore  each  beam 
will  produce  in  the  focal  plane  of  the  lens  LL  a  diffraction 
pattern  like  that  of  figure  77.     These  two  patterns  will  not 
fall  in  exactly  the  same  place.     In  fact  their  centers  will  sub- 
tend, from  the  optical  center  of  the  lens,  an  angle  a,  the  same 
as  that  between  the  two  wavefronts  before  they  reached  the 
lens.     Instead  of  two  sharp  spectrum  lines,  then,  we  have  two 
bands  of  a  certain  width,  each  flanked  by  a  number  of  fainter 
bands,  and  this  would  be  true  even  if  the  slit  of  the  spectro- 
scope were  infinitely  narrow,  and  each  beam  of  light  absolutely 
monochromatic.    The  wider  the  beam  of  light,  the  more  nearly 
these  bands  would  approximate  to  real  lines.     If  the  angle  a 
is  quite  small,  and  the  width  of  the  beams  is  also  small,  the 
two  central  bands  will  run  together,  so  that  only  a  single  band 
will  be  seen,  no  matter  how  much  we  try  to  magnify  the  image 
with   a  high-power   eyepiece.      The   question   then   arises,   how 
small  may  be  the  angle  between  the  wavefronts,  a,   in  order 
that  the  two  lines  may  be  resolved,  that  is,  seen  as  distinct.     It 
is  difficult  to  say  exactly  when  the  two  images  run  together, 
but  for  purposes  of  comparison  between  different  instruments 
we  say  that  the  limit  of  resolution  is  reached  when  the  central 
maximum  for  one  image  comes  just  where  the  first  minimum 
of  the  other  image  occurs.    We  have  seen  in  the  preceding  sec- 
tion that  the  angle  between  the  center  of  a  diffraction  pattern 


RESOLVING  POWER  149 

and  the  first  minimum  is  given  by  sin.  $  =  A/cd,  or,  since  0 
is  so  small  that  sin.  0=0  very  approximately,  we  may  write 
0  =  A/cd.  Therefore,  for  resolution  of  the  spectrum  lines  we 
must  have 

a  5  A/cd 

The  lines  will  therefore  be  resolved  only  when  the  angle  between 
the  wavefronts  is  at  least  as  great  as  an  angle  whose  are  is 
equal  to  the  wavelength,  the  radius  being  equal  to  the  width 
of  the  beam.  In  order  that  a  spectroscope  may  have  high 
resolving -power,  it  is  therefore  not  sufficient  that  the  prism  or 
grating  shall  produce  a  great  dispersion  of  the  different  wave- 
lengths. It  is  also  necessary  that  prism  or  grating  shall  be 
wide  enough  to  transmit  a  wide  beam  of  light,  and  the  lenses 
must  also  be  wide  enough  not  to  cut  this  beam  at  the  corners. 
The  same  sort  of  problem  comes  up  in  the  use  of  tele- 
scopes, though  there  is  a  slight  difference  due  to  the  fact  that 
the  beam  of  light  has  then  a  round  cross-section.  It  is  evident 
that  when  a  telescope  is  pointed  toward  a  star  everything  is 
symmetrical  about  the  axis,  and  therefore  the  diffraction  pat- 
tern, instead  of  being  a  central  band  flanked  by  parallel  fainter 
bands,  will  be  a  central  disc,  surrounded  by  a  system  of  faint 
rings.  It  corresponds  very  closely  indeed  to  what  would  result 
if  figure  77  were  revolved  about  its  axis  of  symmetry,  except 
that,  because  the  opening  is  round  and  not  rectangular,  the 
actual  diameters  of  the  dark  rings  are  somewhat  enlarged. 
The  radius  of  the  first  dark  ring  subtends  from  the  optical 
center  of  the  objective  an  angle  whose  value  is  1.22A/cd  instead 
of  A/cd,  if  cd  represents  the  width  of  the  beam  of  light,  that 
is  the  diameter  of  the  objective.  If  the  telescope  be  pointed 
toward  a  close  double-star,  each  star  will  form  in  the  focal 
plane  an  image  of  itself  which  consists  of  a  central  disc  sur- 
rounded by  faint  rings,  and  we  say  that  the  images  are  just 
resolvable  when  the  center  of  one  disc  falls  on  the  first  dark 
ring  of  the  other  diffraction  pattern.  A  double-star  is  resolv- 
able then  if  the  two  components  make  an  angle  which  is  equal 
to  or  greater"  than  1.22A  divided  by  the  diameter  of  the  objec- 
tive. 


150  LIGHT 

A  telescope  of  large  diameter  is  said  to  have  a  large  re- 
solving-power. Similar  considerations  apply  to  a  microscope, 
though  the  treatment  is  somewhat  different  because  with  that 
instrument  the  light  does  not  enter  the  objective  in  plane  waves 
but  in  a  highly  diverging  beam.  All  optical  images  are  affected 
by  diffraction,  and  this  explains  the  statement  made  in  section 
38  that  no  lens,  however  perfect  in  design  and  workmanship, 
can  produce  a  point  image  from  a  point  source. 

The  question  is  often  asked,  by  those  unacquainted  with 
optical  instruments, — "How  small  may  an  object  be,  or  how 
far  away  may  it  be,  and  yet  be  visible  in  a  given  microscope 
or  telescope  ? "  Such  a  question  is  not  pertinent,  because  any 
microscope  will  render  visible  any  object,  however  small,  and 
any  telescope  will  reveal  any  object  however  small  or  remote, 
provided  that  object  gives  out  'a  sufficiently  strong  beam  of 
light.  The  proper  question  is, — "How  close  together  (in  linear 
measure  for  a  microscope,  in  angular  measure  for  a  telescope) 
may  two  small  objects  be  and  yet  be  seen  as  distinct  and  sep- 
arate in  the  instrument!"  The  efficiency  of  a  telescope  in  this 
respect  depends  entirely  upon  the  diameter  of  its  objective 
lens,  not  at  all  upon  its  focal  length,  consequently  the  diameter 
is  by  far  the  more  important  dimension. 

The  eye,  like  any  other  optical  instrument,  has  a  definite 
limit  to  its  resolving-power,  depending  upon  the  diameter  of 
the  pupil.  Anyone  can  convince  himself  of  the  truth  of  this 
statement  by  the  following  simple  experiment :  Make  two  clear 
dots  upon  a  blackboard,  about  %  inch  apart,  and  then  walk 
backward  away  from  the  wall  with  the  eyes  fixed  upon  the  dots. 
At  a  certain  distance,  the  two  seem  to  run  together,  and  can 
no  longer  be  distinguished  as  separate  dots.  Strange  as  it 
may  seem,  it  is  possible  that  they  may  be  distinguished  at  a 
greater  distance  in  a  relatively  dark  than  in  a  very  bright 
room,  for  in  very  bright  light  the  pupil  contracts,  and  the 
resolving-power  is  decreased. 

The  diffraction-rings  produced  in  the  eye  are  finer  in 
structure  than  the  structure  of  the  retina  itself,  if  the  pupil 
is  normally  open,  ,and  therefore  are  not  ordinarily  visible,  but 
they  become  visible  if  an  effective  decrease  in  the  diameter  of 
the  pupil  is  made  artificially  by  looking  through  a  very  small 


RESOLVING  POWER  151 

hole  in  a  card  or  other  opaque  object.  The  following  experi- 
ment is  instructive:  Make  a  single  straight  cut  with  a  knife 
in  a  stiff  card,  so  as  to  make  a  sort  of  slit.  Hold  this  closie  to 
the  eye  and  look  through  it  at  a  single  filament  of  an  incan- 
descent lamp,  keeping  the  slit  parallel  to  the  filament.  The 
diffraction  bands  described  in  connection  with  figures  76  and 
77  will  be  clearly  seen.  It  will  be  noticed  that  the  bands 
become  narrowed  if  the  card  is  sprung  so  that  the  opening  is 
widened,  and  conversely  if  the  opening  is  narrowed.  Here  the 
lens  of  the  eye  takes  the  place  of  the  lens  LL  in  figure  76,  the 
opening  in  the  card  that  of  the  opening  cd,  and  the  retina  that 
of  the  screen  AB.  Notice  that  the  central  maximum  is  50% 
farther  from  either  of  the  secondary  maxima  on  either  side, 
than  any  two  adjacent  secondary  maxima  are  from  one  another. 

Problems. 

1.  A  certain  prism  of  dense  flint  glass  separates  the  two 
yellow  sodium  wavelengths  by  an  angle  of  2  seconds  of  arc. 
How  wide  must  the  beam  be,  in  order  that  the  two  lines  may 
be  seen  separated  in  the  spectroscope? 

2.  A  "double-star"  has  components  whose  angular  separa- 
tion is  only  0.16  second  of  arc.     What  is  the  diameter  «f  the 
objective  of  the  smallest  telescope  that  can  resolve  them? 

3.  Find  the  resolving-power  of  the  eye,  when  the  pupil 
is        inch  in  diameter. 


CHAPTER  IX. 

74.  Young's  interference  experiment. — 75.  The  biprism. — 76.  Inter- 
ference in  thin  uniform  films. — 77.  Change  of  phase  on  reflection. — 78. 
Non-uniform  films. — 79.  The  Michelson  interferometer. — 80.  Newton's 

rings. gl.  Fabry  and  Perot  interferometer. — 82.  Interference  in  white 

light.— 83.  Rainbows.— 84.  Motion  relative  to  the  ether.— 85.   The  re- 
lativity theory. 

74.  Young's  interference  experiment.— The  arrangement 
of  mirrors,  invented  by  Fresnel  to  show    the    interference    of 
light,  and  described  in  section  22   (see  figure  21),  is  not  the 
only  device  for  the  purpose,  though  perhaps  the  most  satis- 
factory.    The  earliest  was  due  to  Thomas  Young,  one  of  the 
early  champions  of    the    wave    theory.     It    consists    of    a  slit 

through  which  light  streams  to 
an  opaque  screen  containing  two 
very  small  holes  close  together. 
A  white  screen  farther  on  re- 
ceives the  light  from  the  two 
holes,  which  could  be  replaced 
with  advantage  by  two  fine  slits 
Figure  78  parallel  to  the  original  slit. 

Figure  78  shows  this  simple  arrangement.  S  is  the  source  slit, 
X  and  Y  the  two  round  holes  or  small  slits.  These  last  are 
small  enough  so  that  diffraction  causes  the  light  coming  through 
them  to  spread  out.  Light  from  the  two  holes  will  therefore 
overlap  in  the  neighborhood  of  C  on  the  white  screen  AB,  and 
will  produce  interference  fringes  there.  The  student  will 
observe  that  the  opaque  screen  with  two  slits  through  it  is  to 
all  intents  and  purposes  a  very  coarsely  ruled  grating  with 
only  two  openings.  Although  this  arrangement  is  easy  to  con- 
struct, very  little  light  comes  through  the  two  slits,  and  the 
fringes  are  therefore  very  dim  indeed. 

75.  The  biprism. — This  is  another  device  of  Fresnel.    It 
consists  of  two  identical  glass    prisms,    of    very    small    angle, 
cemented  together  base  to  base.     Figure  79  shows  how  it  is  set 
up.    B  is  the  biprism,  S  the  slit-source,  ab  a  white  screen.    The 

(152) 


INTERFERENCE  IN  THIN  FILMS 


153 


light  passing  through  the  upper  half  of  the    biprism    is    bent 
downward,  that  through  the  lower  half  upward,  so  that  the 

a  light  comes  to  the  screen  as 
if  it  came  from  two  points 
X  and  Y,  which  may  be  re- 
garded  as  virtual  images  of 
the  real  slit  S>  formed  by  re- 
fraction  through  the  upper 
and  lower  halves  of  the  bi- 
prism respectively.  Where 
the  two  beams  overlap  near 
Fi«ure  79  the  center  of  the  white 

screen,  interference  fringes  are  produced. 

76.  Interference  in  thin  uniform  films. — Nature  herself 
provides  us  with  the  finest  cases  of  interference,  in  the  beauti- 
ful iridescent  coloring  of  soap-films,  films  of  oil  on  water,  fis- 
sures in  the  interior  of  crystals,  etc.  In  all  such  cases,  it  is 
found  that  there  are  two  reflecting  surfaces  separated  by  a 
very  small  distance,  and  the  interference  is  between  the  light 
reflected  from  the  first  surface  and  that  reflected  from  the 
second.  It  is  further  necessary,  ,if  the  colors  are  to  be  seen 
anywhere  but  in  certain  particular  spots  when  the  eye  happens 
to  be  in  the  correct  position,  that  the  light  shall  come  from  a 
widely  extended  source,  such  as  the  sky,  the  whitened  wall  of 


Figure 


a  room,  or  a  broad  flame,  giving  out   light    in    all    directions 
from  every  point  of  it. 

We  shall  first  suppose  that  the  film  has  plane  and  parallel 
surfaces,  so  that  it  is  of  uniform  thickness.  Let  AB  and  CD, 
figure  80,  represent  the  upper  and  lower  surfaces  of  suoii  a 


154  LIGHT 

film,  and  E  the  position  of  the  eye.  Let  t  be  the  thickness  of 
the  film,  and  n  its  index  of  refraction.  At  first  we  shall  sup- 
pose that  the  extended  radiating  surface,  not  shown  in  the 
diagram,  but  well  above  the  position  E,  gives  light  of  only  a 
single  wavelength  A.  If  only  the  upper  surface  of  the  film 
reflected  light,  the  eye  could  look  in  any  direction,  such  as  EX, 
and  see  part  of  the  extended  source  reflected  in  this  surface, 
the  course  of  a  ray  of  light  being  given  by  the  lines  SX  and 
XE.  But  only  part  of  the  light  is  reflected  at  the  first  surface, 
while  the  rest  enters  the  film  along  the  ray  XK,  a,iid  part  of 
it  is  reflected  at  the  second  surface  along  the  ray  KZ.  When 
the  ray  KZ  strikes  the  upper  surface,  part  of  it  is  again  re- 
flected, but  part  is  refracted  out  and  follows  a  ray  parallel 
to  the  ray  first  reflected,  XE.  There  will  in  fact  be  still 
another  ray  which  will  emerge  parallel  to  XE  after  three 
reflections,  another  after  five,  and  so  on,  each  one,  however, 
weaker  than  the  rays  that  have  undergone  fewer  reflections. 
Since  the  film  must  be  very  thin  to  show  interference,  all 
these  rays  will  lie  very  close  together,  and  may  therefore  enter 
the  pupil  of  the  eye  together.  If  the  eye  is  focussed  for 
infinitely  distant  objects,  that  is  for  parallel  rays?  they  will  all 
be  brought  to  focus  at  the  same  point  on  the  retina. 

Let  us  first  see  what  difference  in  phase  exists,  at  the 
retina,  between  light  in  the  ray  XE  and  that  in  the  parallel 
ray  from  Z,  which  has  suffered  only  one  reflection  within  »the 
film.  Draw  YZ  perpendicular  to  the  two  rays.  From  Y  and 
Z  on,  there  are  the  same  number  of  wavelengths  in  each  path, 
since  a  wavefront  such  as  YZ  would  be  focussed  upon  the 
retina.  Also  draw  XP  perpendicular  to  KZ.  A  wavefront  in 
the  position  XP  would  be  reflected  to  the  position  YZ  011 
emerging  into  the  upper  medium.  Therefore  there  are  the 
same  number  of  wavelengths  in  XY  and  ZP,  and  the  path- 
difference  between  the  two  rays  is  simply  the  distance  XK 
+  KP,  in  the  medium  of  refractive  index  n.  Draw  XP  per- 
pendicular to  the  surfaces  AB  and  CD,  and  prolong  it  till  it 
meets  ZK  produced  at  G.  The  right-angled  triangles  XFK 
and  GFK^are  equal,  therefore  GK  —  XK,  and  the  path-dif- 
ference XK  +  KP  —  GP.  The  angle  at  G  is  the  angle  of 
refraction,  r,  and  XG  =  2t.  Therefore  the  path-difference  is 


INTERFERENCE  IN  THIN  FILMS  155 

2t.  cos  r.  Since  the  wavelength  within  the  film  is  A/n,  the 
number  of  wavelengths  in  the  path-difference  is 

A       '2nt.  cos  r 
2t.  cos  r  ^-   —  —  - 

n  A 

77.  Change  of  phase  on  reflection. — But  difference  in  path 
is  not  the  only  difference  in  the  two  rays  SXY  and  SXKZ, 
for  they  have  suffered  reflections  of  quite  different  kinds. 
While  the  reflection  at  X,  occurs  011  the  rarer  side  of  the  sur- 
face AB,  that  at  K  occurs  on  the  denser  side  of  the  surface 
CD.  and  it  is  an  important  fact  that  a  reflection  on  the  rarer 
side  of  a  boundary  is  always  accompanied  by  an  abrupt  change 
of  phase  which  is  equivalent  to  the  introduction  of  half  a 
wavelength  in  the  path.  That  is,  a  crest  is  reflected  as  a  trough, 
and  vice  versa.  No  such  change  of  phase  exists  when  the 
reflection  occurs  on  the  denser  side  of  the  boundary.  Crest  is 
reflected  as  crest  and  trough  as  trough.  An  actual  proof  of 
these  statements  involves  difficult  mathematics,  but  their  rea- 
sonableness may  be  made  plain  by  a  consideration  of  the  much 
simpler  case  of  waves  running  along  a  string.  In  figure  81 
XY  represents  a  light 
string,  joined  at  Y  to 
the  end  of  a-  much 
heavier  string  YZ. 
Since  transverse  waves  Figure  si 

travel  faster  along  a  light  than  along  a  heavy  string,  the  light 
string  takes  the  place  of  the  air  in  figure  80,  the  heavy  string 
that  of  the  film,  and  the  point  Y  represents  one  of  the  bounding 
surfaces.  In  the  upper  diagram  of  figure  81  a  single  crest  A  is 
i  shown  advancing  toward  the  boundary  point  Y.  Suppose  for  a 
moment  that  the  end  Y  were  clamped  tightly.  Then,  when  the 
crest  A  struck  it,  the  reaction  at  this  fixed  point  would  throw 
the  cord  below  the  normal  position,  and  a  reflected  trough 
would  start  back  toward  the  left.  With  the  end  Y  not  fixed 
rigidly,  but  merely  fastened  to  the  heavier  cord,  a  trough  is 
still  reflected  to  the  left  along  the  lighter  cord,  while  a  crest 
goes  on  along  the  heavier  one.  There  is  a  certain  analogy  here 
with  the  case  of  a  light  ball  striking  squarely  a  heavier  one 
which  was  originally  stationary,  both  being  perfectly  elastic. 


156  LIGHT 

The  direction  of  motion  of  the  striking  ball  is  reversed,   and 
part  of  its  motion  is  given  up  to  the  other. 

If  the  ball  originally  in  motion  is  the  heavier,  its  direction 
of  motion  is  not  reversed  upon  striking  the  lighter  one,  but 
only  somewhat  checked,  both  balls  moving  off  in  the  same 
direction  after  impact.  The  analogue  to  this  case  is  provided 
when  a  crest  approaches  the  point  Y  along  the  heavier  string, 
as  indicated  by  A  in  the  lower  diagram  of  figure  81.  Both  the 
reflected  and  the  continuing  waves  will  then  be  crests. 

'  Of  course,  in  either  case,  the  reflected  and  the  continuing 
waves  will  both  have  amplitudes  less  than  that  of  the  incident 
wave.  In  fact  the  energy  of  the  incident  wave  is  divided 
between  the  two. 

Let  us  apply  these  principles  to  the  optical  case  of  figure 
SO.  The  incident  wave  coming  along  from  S,  like  that  along 
the  lighter  string  in  the  upper  diagram  of  figure  81,  will  give 
rise  to  a  reflected  wave  of  reversed  phase  travelling  toward  E 
and  a  continuing  (refracted)  wave  without  reversal  of  phase 
travelling  toward  K.  These  two  will  divide  between  them  the 
energy  of  the  original  wave.  On  the  other  hand,  the  wave 
coming  from  X  to  K,  like  that  in  the  heavier  string  in  the 
lower  diagram  of  figure  81,  will  produce  a  reflected  wave  trav- 
elling toward  Z  and  a  continuing  (refracted)^  wave  travelling 
toward  T,  neither  of  which  has  its  phase  reversed. 

Return  now  to  consideration  of  the  difference  in  phase 
between  the  ray  reflected  at  the  upper  surface  of  the  film,  and 
the  one  which  has  undergone  one  reflection  inside  the  film. 
We  have  found  that  the  difference  in  path,  expressed  in  wave- 
lengths, is 

2nt.  cos  r 

X 
To  this  must  be  added  the  equivalent  of   one   half   wavelength 

on  account  of  the  dissimilarities  in  the  two  cases  of  reflection. 
Therefore,  interference  will  occur  when 

2at.cosr  I     JL^N-K- 

or  when 

2nt.  cos  r     _  „ 

A 
N  being  any  whole  number. 


INTERFERENCE  IN  THIN  FILMS  157 

As  a  matter  of  fact,  the  ray  from  Z  is  considerably  weaker 
than  that  from  X,  and  therefore  cannot  completely  neutralize 
it.  But  if  we  consider  the  ray  from  U,  which  has  undergone 
three  internal  reflections,  we  see  that  its  path  exceeds  that 
from  Z  by  exactly  the  amount  by  which  the  latter  exceeds  that 
from  X,  viz.,  in  wavelengths, 

2nt.  cos  r 

A 
but  that  there  is  no  occasion  in  either  of  the  rays  from  Z  or 

U  for  the  change  of  phase  on  reflection.  Consequently  the  rays 
from  U  and  Z  will  differ  by  a  whole  number  of  wavelengths  in 
phase,  and  therefore  will  be  in  condition  to  assist  one  another, 
when  the  rays  from  Z  and  X  are  opposite  in  phase,  so  as  to 
interfere.  In  the  same  way  the  rays  that  have  undergone  5, 
7,  or  any  odd  number  of  internal  reflections,  will  also  be  in 
phase  with  that  from  Z.  It  can  be  shown  that  the  sum  of  the 
amplitudes  of  the  rays  that  have  undergone  1,  3,  5,  7,  etc., 
internal  reflections  is  just  equal  to  the  amplitude  of  the  ray 
that  has  had  only  one  external  reflection.  Consequently,  when 
the  thickness  t,  the  angle  r,  and  the  wavelength  A  are  such  that 

2nt.  cos  r 

A 

a  whole  number,  the  eye  will  see  no  light  whatever  in  this 
direction.* 

*There  are  certain  considerations  of  continuity  which  would  lead 
us  to  believe,  quite  apart  from  the  demonstration  of  figure  81,  that  a 
change  of  phase  must  be  caused  by  either  internal  or  external  reflec- 
tion, and  not  by  both.  As  a  film  becomes  thinner  and  thinner,  it 
should  approach,  in  optical  qualities,  the  condition  of  no  film  at  all; 
in  other  words,  a  film  whose  thickness  is  very  small  compared  to  the 
wavelength  of  light  should  fail  completely  to  reflect  light,  allowing  it 
to  pass  through  unimpeded.  In  fact,  it  is  easy  to  test  this  point,  for  a 
soap-film  that  is  allowed  to  drain  and  evaporate  becomes  in  some  places 
much  thinner  than  the  wavelength  of  visible  light.  Such  places  are 
known  as  "black  spots,"  and  they  have  the  appearance  of  irregularly 
shaped  holes  through  the  film,  although  the  fact  that  the  whole  film 
does  not  collapse  shows  there  is  not  a  real  hole.  (Incidentally,  this 
experiment  indicates  that  the  diameter  of  a  molecule  must  be  much 
smaller  than  the  wavelength  of  visible  light). 

But,  if  there  were  no  difference  in  phase  introduced  by  the  two 
kinds  of  reflection,  the  only  cause  of  phase  difference  would  be  dif- 


158  LIGHT 

Now  since,  the  thickness  of  the  film  being  constant,  the 
condition  for  interference  varies  only  with  the  angle  of  refrac- 
tion, or — what  comes  to  the  same  thing — with  the  angle  of 
incidence,  then  if  the  eye  observes  darkness  in  any  such  direc- 
tion as  EX  it  will  also  observe  darkness  in  any  direction,  such 
as  EV,  for  which  the  angle  of  incidence  is  the  same.  That  is, 
it  will  observe  a  dark  ring,  subtending  a  cone  whose  angle  is 
XEV.  Since  darkness  occurs  whenever 

2nt.  cosr 

A 

where  N  may  have  any  one  of  the  integral  values,  0,  1,  2,  3, 
etc.,  there  will  be  a  series  of  such  rings.  It  has  already  been 
intimated  that  the  eye  must  be  focussed  for  parallel  rays,  as 
if  looking  at  an  infinitely  distant  object,  in  order  that  these 
rings  may  be  seen.  There  is  also  another  reason  why  they 
look  as  if  they  were  very  distant:  since  the  appearance  of  a 
dark  ring  depends  only  upon  the  angle  of  incidence,  and  not 
at  all  upon  the  position  of  the  eye,  if  the  latter  be  moved  the 
rings  will  move  with  it,  instead  of  appearing  fixed  within  the 
film.  In  fact,  they  seem  to  be  seen  through  the  film,  just  as 
one  sees  distant  clouds  or  landscape  through  a  window.  For 
these  reasons,  it  is  said  that  the  rings  of  interference  seen  in 
thin  films  of  uniform  thickness  are  "located  at  infinity." 

The  ring  for  which  N  —  1  is  called  the  ring  of  first  order, 
that  for  which  N  =  2  the  ring  of  second  order }  etc.  Of  course, 
the  dark  rings  are  separated  by  bright  rings  where  the  light 
reflected  from  the  upper  and  lower  surfaces  are  more  or  less 
in  phase. 

If  white  light  instead  of  monochromatic  falls  upon  the 
film,  a  series  of  colored  rings  will  be  seen,  instead  of  merely 
dark  rings  separated  by  bright  rings  all  of  one  color.  In  a 
real  soap-film  the  colors  are  of  irregular  outline  instead  of  in  a 
ring  pattern,  because  the  film  is  never  of  uniform  thickness 
and  the  surfaces  are  usually  not  plane.  Where  the  angle  of 

ference  in  path,  and  for  vanishing  thickness  of  the  film  all  the  reflect- 
ed beams  would  have  the  same  length  of  path.  Th'us,  experiment,  as 
well  as  general  reasoning  from  the  principle  of  continuity,  indicate  an 
abrupt  change  in  the  phase  of  one  of  the  reflected  rays. 


NON-UNIFORM  FILMS  159 

incidence  and  the  thickness  are  such  as  to  produce  interference 
for  light  of  a  certain  wavelength,  other  wavelengths  are  re- 
flected without  interference,  and  thus  the  colors  are  produced. 

78.  Non-uniform  films. — When  a  film  is  not  of  uniform 
thickness,  its  two  faces  will  not  be  parallel,  and  the  theory  of 
the  interference  becomes  much  more  difficult.  One  thing  about 
the  interference  bands  produced  in  such  a  case,  however,  is  in 
striking  contrast  with  those  for  films  of  uniform  thickness, 
viz.,  the  fact  that  they  are  not  located  at  infinity,  but  seem  to 
lie  in,  or  very  close  to,  the  film  itself.  Figure  82  will  show  a 
reason  for  this  fact.  AB  and  CD 
are  the  two  surfaces  of  a  non- 
uniform  film,  Sx  an  incident  ray, 
xE  a  ray  reflected  from  the  upper 
surface,  and  zE'  a  ray  emerging 
after  one  reflection  inside  the  film.  Figure  82 

Under     these     circumstances,     xE 

and  zE'  are  not  parallel.  In  order  that  both,  after  entering  the 
pupil  of  the  eye,  shall  be  brought  to  focus  at  the  same  point  of 
the  retina,  the  eye  must  be  focussed,  not  upon  infinity,  but  upon 
the  point  where  the  two  rays  cross,  which  is  either  within  the 
film  or  close  to  it.  The  difference  in  phase  varies  with  both  the 
angle  of  incidence  and  the  thickness  of  the  film,  but  principally 
with  the  latter.  Consequently  a  single  dark  fringe  seen  on  the 
surface  maps  out  very  closely  the  points  of  equal  thickness. 

Suppose  one  sheet  of  plate  glass  be  laid  upon  another,  and 
viewed  by  the  reflected  light  of  a  sodium  burner.  A  very  thin 
film  of  air  is  left  between  the  plates,  which  is  usually  quite 
far  from  uniform  in  thickness,  because  the  surface  of  commer- 
cial plate  glass  is  never  really  plane.  Regions  of  equal  thick- 
ness of  the  air  film  are  mapped  out  by  curved  bright  and  dark 
lines.  Suppose  that  at  a  certain  fringe  the  thickness  of  the 
film  is  5A.  Therr,  along  the  next  fringe  on  one  side  it  is  5.5A, 
along  the  next  on  the  other  side  it  is  4.5x.  The  thickness  in- 
creases by  half  a  wavelength  from  fringe  to  fringe,  because 
the  light  reflected  from  the  lower  surface  of  the  film  traverses 
the  film  twice.  If  one  of  the  glass  surfaces  is  known  to  be 
plane,  the  method  of  fringes  in  reflected  light  may  be  used  to 


160  LIGHT 

test  the  other  for  planeness,  and  to  show  what  parts  must  be 
ground  down. 

79.  The  Michelson  interferometer, — A  very  valuable  in- 
strument, whose  action  depends  upon  the  same  principle  as 
interference  in  thin  films,  is  the  Michelson  interferometer, 
shown  diagrammatically  in  figure  83.  AB  and  CD  are  plane 
glass  mirrors,  heavily  silvered  011  their  front  surfaces,  the 
former  fixed  in  position,  the  latter  so  mounted  that  it  can  be 
moved  backward  and  forward,  in  a  direction  perpendicular  to 
its  own  plane,  by  a  fine-pitched  screw.  EF  is  another  plane 

glass  mirror,  set  at  an  angle 
of  45°,  and  fixed  in  position. 
On  its  upper  right-hand  sur- 
face it  is  covered  with  a  very 
thin  coat  of  silver,  so  that  it 
will  reflect  about  half  of  all 
light  falling  upon  it  and 
>  - ,  allow  about  half  to  pass 


\>  through,  that  is,  it  is  a  half- 

silvered  mirror.  An  extended 
source    of   monochromatic 
light,     such     as     a     sodium 
Figure  83  flame,    (Bunsen    burner    fed 

with  common  salt)  is  placed  at  L.  A  small  part  of  the  light  is 
reflected  from  the  lower  unsilvered  surface  of  the  mirror  EF, 
but  this  is  weak  enough  to  be  ignored.  Of  that  which  reaches 
the  half-silvered  surface,  half  passes  through  to  the  fixed  mir- 
ror AB,  is  there  reflected  back  to  EF  and  is  again  half  re- 
flected, to  the  eye  placed  at  I.  The  other  part  of  the  light! 
which,  coming  from  L,  strikes  the  half-silvered  surface,  is 
reflected  to  CD  and  back  again,  and  half  of  it  passes  through 
EF  to  the  eye.  The  difference  in  path  between  these  two 
beams  causes  interference  fringes  to  be  seen.  The  fourth  glass 
GH  is  added  because  without  it  one  of  the  interfering  beams 
would  pass  through  three  thicknesses  of  glass,  the  other  only 
through  one.  GH  must  be  of  the  same  thickness  as  EF  and 
parallel  to  it,  but  it  is  not  silvered. 

So  far  as  the  eye  is  concerned,  we  may  regard  the  fixed 
mirror  AB  as  replaced  by  its  image,  seen  by  reflection  in  EF, 
which  would  come  in  some  such  plane  as  ab.  If  AB  be  ad- 


MIOHELSON  INTERFEROMETER  161 

justed,  by  means  of  screws  provided  for  that  purpose,  so  that 
its  image  is  accurately  parallel  to  CD,  we  shall  have  in  effect 
light  reflected  from  two  plane  and  parallel  surfaces,  ab  and 
CD,  just  as  in  figure  80,  with  the  single  exception  that  when 
we  are  dealing  with  a  real  film,  as  in  figure  80,  multiple  reflec- 
tions occur  within  the  film,  causing  a  series  of  reflected  beams, 
while  here  we  have  only  two.  The  interference  fringes  will 
therefore  be  a  system  of  concentric  circular  rings,  apparently 
located  at  an  infinite  distance. 

Any  displacement  of  the  movable  mirror  CD  will  change 
the  thickness  of  the  hypothetical  airfilm  between  CD  and  ab, 
and  cause  a  corresponding  change  in  the  rings,  either  an 
increase  or  a  decrease  in  their  diameter.  In  order  to  determine 
whether  an  increase  in  the  distance  ab  to  CD  causes  the  rings 
to  enlarge  or  diminish,  let  us  fix  our  attention  on  a  certain 
dark  ring,  say  of  order  200.  We  have  seen  that  the  difference 
in  path,  expressed  in  wavelengths,  will  be 

2nt.  cos  r      XT  . 

-  =  N 
A 

Here,  n  =  1,  because  the  film  is  air,  and  therefore  the  angles 
of  refraction  and  incidence  are  equal.  Also,  N  =  200.  There- 
fore 


A 

t  is  of  course  the  distance  ab  to  CD.  If  t  increases,  the  factor 
cos  i  must  decrease  by  the  same  amount,  in  order  that  the  ring 
of  order  200  shall  still  be  seen.  But  a  decreasing  cosine  means 
an  increasing  angle,  and  therefore,  for  greater  thickness  of  the 
airfilm,  one  must  look  more  obliquely  in  order  to  see  this  ring, 
that  is  the  ring  enlarges.  All  the  rings  enlarge  as  CD  moves 
away  from  ab,  and  spots  appear  in  the  center  one  after  another 
and  expand  to  form  new  rings.  Conversely,  if  CD  moves 
toward  ab.  the  rings  all  decrease,  and  one  after  another  they 
shrink  to  mere  spots  in  the  center  and  disappear. 

The  appearance  or  disappearance  of  each  bright  spot  at 
the  center  indicates  that  another  half  wavelength  has  been 
added  to  or  subtracted  from  the  thickness  of  the  airfilm.  For 
in  the  center  the  incidence  is  normal,  that  is,  i  =  0,  and  the 
path-difference  becomes  merely  2t.  It  is  therefore  easy  to  de- 


162  LIGHT 

termine  accurately  how  far,  in  terms  of  light  wavelengths,  the 
mirror  CD  is  moved,  by  simply  counting  the  number  of  rings 
that  appear  or  disappear  at  the  center  and  dividing  by  2. 
Professor  Michelson  has  used  this  instrument  to  find  the  length 
of  the  standard  international  meter,  in  terms  of  the  wavelength 
of  a  red  line  in  the  spectrum  of  cadmium.  The  actual  details 
of  the  experiment  were  complicated  and  laborious,  but  the 
degree  of  accuracy  obtained  was  marvelous.  A  very  readable 
account  of  this  and  of  other  scientific  uses  of  the  interfero- 
meter will  be  found  in  Michelson 's  book  "  Light  Waves  and 
their  Uses." 

If  the  fixed  mirror  AB  is  so  adjusted  that  its  image  ab  is 
not  parallel,  but  slightly  inclined  to  CD,  we  shall  have  the  case 
of  a  non-uniform  film,  and  the  bands,  instead  of  being  circular, 
will  be  nearly  straight,  mapping  out  regions  of  approximately 
equal  thickness.  Instead  of  being  located  at  infinity,  they  will 
seem  to  lie  near  ab  and  CD.  A  movement  of  CD  causes  them 
to  move  across  the  faces  of  the  mirrors. 

80.  Newton's  rings. — The  interference  phenomenon  known 
as  "Newton's  rings"  is  a  matter  of  considerable  historical  inter- 
est. As  its  name  indicates,  it  was  known  to  Sir  Isaac  Newton. 
It  is  clearly  an  example  of  interference  in  an  airfilm  whose 
thickness  is  not  uniform,  but  Newton,  refusing  to  entertain  the 
wave  theory,  gave  a  different  and  somewhat  cumbersome  ex- 
planation which  now  has  no  interest  for  us. 

These  rings  are  produced  by  placing  upon  a  plane  piece 
of  glass  a  plano-convex  lens  of  very  slight  curvature,  convex 
side  in  contact  with  the  plane  glass.  Between  the  two  pieces 
there  is  a  film  of  air,  whose  thickness  varies  from  zero  in  the 
middle,  at  the  point  of  contact,  to  a  great  many  wavelengths 
at  the  edges.  Figure  84A  is  a  photograph  of  this  arrangement 
as  it  appears  when  illuminated  from  the  front  by  monochro- 
matic light.  In  this  photograph  the  light  was  of  violet  color, 
and  was  obtained  by  separating  the  light  from  a  mercury-are 
into  its  spectrum.  The  fringes  appear  somewhat  elliptical  in  the 
figure,  because  the  photograph  was  necessarily  taken  somewhat 
obliquely.  They  are  in  fact  circular,  marking  regions  of  equal 
thickness.  The  center,  where  the  two  glasses  come  into  con- 
tact, is  always  black,  corresponding  to  zero  difference  in  path, 


FABRY  AND  PEROT  INTERFEROMETER         163 

unless  dust  or  some  other  obstruction  prevents  actual  contact. 
Figure  84B  is  the  same  thing  when  illuminated  by  white  light, 
instead  of  monochromatic.  The  difference  will  be  explained 
later. 

81.  Fabry  and  Perot  interferometer. — If  the  student  will 
refer  again  to  figure  80,  he  will  readily  understand  that  not 
only  the  reflected  light,    but    also    that   which    is    transmitted 
through  the  film,  should  show  interference  fringes..     For  some 
of  the  light  passes  through  the  film    without    being    reflected, 
some  passes  through  after  two  internal  reflections,  some  after 
four,  etc.,  and  these  different  rays,  all  parallel  on  emergence 
from  the  film,  should  be  in  a  condition  to  interfere  for  certain 
angles  of  incidence.     Such  fringes    are    indeed    seen    in    the 
transmitted  light,  but  they  are  rather  weak,  because  the  ray 
that  passes  through  without  reflections    is    so    much    stronger 
than  any  of  the  others  that  the  interference  is  not  complete. 
Only  faint  brighter  and  darker  rings  on  a  rather  bright  back- 
ground are  seen. 

Fabry  and  Perot  conceived  the  idea  of  diminishing  the 
intensity  of  the  first  ray,  and  increasing  that  of  all  the  others, 
by  lightly  silvering  the  surfaces  of  the  film.  Under  these  cir- 
cumstances, the  contrast  between  bright  and  dark  rings  becomes 
much  greater,  and  besides  the  bright  rings  become  much 
sharper,  something  like  spectral  lines, — that  is,  the  width  of 
the  bright  part  is  much  less  than  that  of  the  dark  part.  On 
this  principle,  they  have  developed  a  type  of  interferometer 
that  is  known  by  their  names,  which  for  some  purposes  is 
superior  to  that  of  Michelson.  Its  essential  parts  are  two  discs 
of  perfectly  plane  glass,  parallel  to  one  another,  one  of  them 
fixed  and  the  other  movable  by  means  of  a  screw  in  a  direction 
perpendicular  to  its  own  plane.  The  two  surfaces  turned 
toward  one  another  are  lightly  silvered,  and  the  layer  of  air 
between  is  the  film  in  which  the  interference  takes  place.  Only 
the  transmitted  light  is  used.  The  thickness  of  the  air-film  is 
altered  by  turning  the  screw.  This  instrument  has  been  much 
used  in  recent  years  for  determining  wavelengths.  It  is  more 
accurate  than  a  grating  for  this  purpose.  The  technique  of  its 
use,  however,  is  somewhat  involved. 

82.  Interference  in  white  light. — In  any  interference  ex- 
periment, it  is  found  to  be  impossible  to  observe  interference 


164 


LIGHT 


fringes  if  white,  or  composite,  light  is  used,  except  when  the 
difference  in  path  amounts  to  only  a  few  wavelengths.  In  most 
experiments,  matters  are  so  arranged  that  in  part  of  the  field 
of  view  the  difference  in  path  of  the  interfering  rays  is  zero 
or  nearly  zero,  while  in  other  parts  it  may  be  many  wave- 
lengths; but  in  some,  as  with  a  thin  film  whose  thickness  is 
uniform  but  amounts  everywhere  to  say  more  than  six  or  eight 
wavelengths,  the  difference  in  path  is  always  relatively  large. 
In  cases  of  the  latter  sort,  fringes  are  absolutely  invisible  with 
white  light,  though  they  may  be  very  clear  with  monochromatic 
light.  In  cases  of  the  former  sort,  white  light  shows  a  few 
fringes,  usually  not  more  than  ten  or  a  dozen,  where  the  differ- 


Figure   84 

ence  in  path  is  quite  small,  but  none  whatever  in  the  rest  of 
the  field.  Consider,  for  example,  Newton's  rings.  Here  the 
thickness  of  the  air-film  ranges  from  zero,  where  the  lens  and 
the  flat  disc  come  into  contact,  to  larger  and  larger  values  as 
we  recede  from  that  point.  Figure  84B  is  a  photograph  of  the 
rings  when  the  light  comes  from  a  tungsten  filament  lamp, 
and  is  therefore  composite  white  light.  It  does  not  reproduce 
the  visual  appearance  quite  faithfully,  because  the  range  of 
wravelengths  for  photographic  sensitivity  is  somewhat  different 
from  that  for  visual  sensitivity,  but  the  general  character  is 


INTERFERENCE  IN  WHITE  LIGHT  165 

the  same.  84A,  on  the  other  hand,  is  taken  with  monochroma- 
tic light,  and  the  difference  is  striking.  In  the  original  photo- 
graph, fine  clear  rings  may  be  counted  clear  out  to  the  edges 
of  the  disc  in  84A,  while  in  84B  only  a  few  can  be  seen.  These 
last  of  course  were  brilliantly  colored  in  the  original  object. 

In  order  to  explain  the  difference,  suppose  that,  by  means 
of  a  lens,  an  image  of  the  ring-system  in  white  light  is  focussed 
upon  the  end  of  the  collimator  of  a  spectroscope,  and  that  the 
slit  can  be  moved  sideways  so  as  to  admit  to  the  spectroscope 
light  from  any  chosen  part  of  the  ring-system.  First,  let  it  be 
placed  to  receive  light  from  a  point  where  the  air  film  is 
.000018cm.  thick.  We  suppose  the  illuminating  beam  of  light 
falls -upon  the  lens  and  disc  perpendicularly,  so  that  the  differ- 
ence in  path  for  the  interfering  beams  is  twice  the  thickness, 
or  ,000036cm.  We  have  seen  that,  because  of  the  reversal  of 
phase  on  reflection  from  the  rarer  side  of  a  boundary,  destruc- 
tive interference  takes  place  when  this  difference  in  path  is  one 
wavelength,  or  any  integral  number  of  wavelengths.  There- 
fore absolutely  no  light  of  wavelengths  .000036cm.,  .000018cm., 
.000012cm.,  etc.,  will  enter  the  spectroscope.  Only  the  first  of 
these  would  be  visible  light,  deep  violet  at  that, — the  rest  are 
far  in  the  ultraviolet  and  have  no  effect  upon  the  visible 
appearance  of  the  fringes.  For  light  of  wavelength  .000072cm., 
(deep  red),  the  difference  in  path  is  A/2,  and  since  the  reversal 
in  phase  of  one  of  the  interfering  beams  is  equivalent  to  an- 
other half  wavelength  very  strong  light  of  this  color  would 
enter  the  spectroscope.  Therefore  the  spectrum  would  be  very 
strong  at  the  red  end,  fading  off  into  absolute  darkness  at  the 
violet  end.  Without  the  use  of  the  spectroscope,  this  part  of 
the  ring-system  would  appear  reddish,  or  rather  orange,  for 
the  presence  of  some  color  of  wavelengths  shorter  than  red 
would  undoubtedly  alter  the  appearance  somewhat  toward  the 
yellow  side. 

Now  let  the  slit  be  shifted  to  receive  light  from  a  place 
where  the  thickness  of  the  film  is  twice  as  great,  .000036cm. 
The  difference  in  path  is  now  .000072cm.,  and  the  wavelengths 
destroyed  by  interference  .000072cm.  (deep  red),  .000036cm. 
(deep  violet),  .000024cm.  (ultraviolet),  .000018cm.  (ultravio- 
let), etc.  The  spectrum  would  then  be  quite  dark  at  both  ends, 
shading  into  brightness  in  the  middle,  about  the  green.  The 


166  LIGHT 

color  at  this  part  of  the  ring-system  as  seen  by  the  unaided 
eye  would  undoubtedly  be  green. 

Change  the  slit  again,  so  as  to  receive  light  from  a  place 
of  thickness  .000072cm.,  path-difference  .000144  cm.  The  wave- 
lengths destroyed  by  interference  would  be  .000144  (infra-red), 
.000072  (deep  red),  .000048  (blue-green),  .000036  (deep  vio- 
let), and  certain  ultraviolet  wavelengths.  The  visible  spectrum 
would  be  dark  at  each  end  and  have  a  dark  band  near  the 
middle,  at  the  blue-green,  but  it  would  be  very  bright  in  the 
yellow  and  also  in  the  blue  of  shorter  wavelength,  the  bril- 
liancy shading  off  into  darkness  at  the  two  ends  and  in  the 
middle.  The  color  as  seen  without  the  spectroscope  would  be 
a  compound  of  strong  yellow  and  strong  deep  blue,  with  some 
orange,  green  and  blue-violet.  It  would  probably  be  nearly 
white,  somewhat  yellowish. 

In  the  first  case,  the  spectrum  has  a  single  dark  band,  at 
the  violet  end, — in  the  second,  a  dark  band  at  each  end, — in  the 
third  one  at  each  end  and  one  in  the  middle.  Of  course,  as  the 
bands  become  more  numerous,  they  also  become  narrower,  for 
there  is  always  a  bright  region  between  two  dark  regions.  In 
each  case,  one  of  the  dark  bands  has  come  at  wavelength 
.000036,  but  that  is  only  because  we  chose  places  where  the 
difference  in  path  was  equal  to  that  wavelength  or  a  multiple 
of  it.  With  the  choice  of  another  point,  it  would  have  been 
found  that  the  deep  violet  was  bright  instead  of  dark.  Since 
an  increase  in  the  thickness  of  the  film  causes  more  dark  bands 
in  the  visible  spectrum,  let  us  see  how  many  there  would  be 
at  a  place  where  the  film  is  considerably  thicker,  say  .000216cm. 
The  path-difference  is  .000432,  and  the  wavelengths  destroyed 
by  interference  are  found  Ipy  dividing  this  length  by  the  sep- 
arate integers,  1,  2.  3,  4,  etc.  They  are 

.000432                           .000072  .000033 

.000216                            .000062  .000031       ultra- 

.000144       infrared       .000054  .000029       violet 

.000108                            .000048       visible  etc. 
.000086                           .000043 
.000039 
.000036 

There  would  then  be  seven  dark  bands  in  the  visible  spectrum, 
with  bright  bands  between.  What  would  be  the  color  of  such 


RAINBOWS  167 

light?  "White  light  with  the  violet  end  of  its  spectrum  sup- 
pressed would  be  reddish,  with  the  red  end  suppressed  bluish, 
with  both  ends  suppressed  greenish,  with  the  middle  alone 
suppressed  purple,  and  so  on.  But  when  a  large  number  of 
wavelengths,  distributed  regularly  throughout  the  'spectrum, 
are  cut  out,  there  is  no  reason  why  the  remaining  light  should 
show  one  color  more  than  another,  since  it  contains  constitu- 
ents from  all  the  spectral  regions  in  the  more  general  sense, 
though  not  light  of  every  particular  wavelength.  Therefore, 
at  such  a  place  the  color  would  be  simply  white,  indistinguish- 
able to  the  eye  alone  from  the  light  with  which  the  two  discs, 
flat  plate  and  lens,  are  illuminated.  Evidently  the  same  con- 
dition holds  over  all  parts  of  the  air-film  where  the  thickness 
is  several  times  the  wavelength  of  red  light  or  greater,  and 
since  all  such  parts  are  white,  they  will  appear  uniformly 
illuminated  and  no  fringes  will  be  distinguishable,  although 
in  monochromatic  light  they  could  be  clearly  seen. 

The  proof  given  above  for  the  case  of  Newton's  rings  will 
evidently  hold  good  just  as  well  for  any  sort  of  interference 
experiment,  so  that  we  never  observe  interference  fringes  with 
white  light  unless  the  difference  in  path  between  the  interfer- 
ing beams  is  not  more  than  a  few  wavelengths. 

83.  Rainbows. — The  colors  of  rainbows  are  caused  partly 
by  interference,  partly  by  a  dispersion  of  light  passing  through 
raindrops,  very  similar  to  the  action  of  a  prism.  The  approxi- 
mate positions  of  the  principal  bows  is  given  by  considering 
only  the  dispersion  by  the  raindrops.  The  modifications  intro- 
duced into  the  theory  by  the  interference  phenomena  are  diffi- 
cult to  understand,  and  they  will  be  omitted  from  the  discus- 
sion here  given. 

We  shall  first  take  up  the  primary  bow,  the  only  one  that 
is  usually  seen.  In  order  to  see  it,  the  back  must  be  turned  to 
the  sun,  and  then  it  appears,  if  there  are  any  raindrops  in  the 
right  position  and  the  direct  sunlight  reaches  them,  as  an  arc, 
whose  center  lies  on  a  prolongation  of  a  line  from  the  sun 
through  the  eye  of  the  observer.  If  the  observer  changes  his 
position,  the  bow  seems,  to  move  with  him,  so  that  the  bow, 
the  sun  and  the  observer  always  keep  the  same  relative  position. 
Since  this  is  what  would  happen  if  the  bow  were  infinitely 
distant,  we  say  that,  like  the  rings  seen  by  reflection  in  a  thin 


168  LIGHT 

uniform  film,  the  bow  is  located  at  infinity.  This  is  true  even 
when  it  is  formed  by  water  drops  quite  close  at  hand,  as  in  the 
spray  from  a  fountain.  The  bow  seems  to  move-  through  the 
fountain  as  the  observer  moves,  instead  of  staying  fixed  in  it. 

The  bow  is  caused  by  light  that  has  suffered  two  refrac- 
tions and  one  reflection:  that  is,  by  light  that  has  been  re- 
fracted into  the  raindrop,  reflected  once  inside,  and  then  re- 
fracted out  again.  Of  course,  the  refractions  cause  a  disper- 
sion of  the  light,  just  as  a  prism  would,  but  the  case  is  less 

simple  than  that  for  a 
prism,  because  the  sur- 
faces of  the  raindrop 
are  curved  while  those 
of  a  prism  are  plane. 
When  parallel  rays  fall 
upon  a  prism,  all  the 
light  of  a  given  wave- 
length will  be  refracted 
into  the  same  direction. 

but  in  the  case  of  the  spherical  drop  they  are  spread  out  into  a 
number  of  different  directions.  This  can  be  seen  from  figure  85, 
where  three  parallel  rays  from  the  sun  are  drawn,  striking  the 
sphere  at  slightly  different  positions,  therefore  having  different 
paths  through  the  drop  and  emerging  in  quite  different  direc- 
tions below.  The  angle  D,  between  the  direction  of  a  ray  before 
entering  the  drop  and  its  direction  after  leaving,  called  the 
deviation  of  the  ray,  is  different  for  rays  striking  the  surface 
at  different  angles  of  incidence.  It  is  not  difficult  to  show  thai 
for  any  ray 

D  =  180°  +  2i  —  4r 

where  i  is  the  angle  of  incidence  and  r  the  corresponding  angle 
of  refraction.  If  we  calculate  and  plot  the  values  of  D  for 
various  values  of  i,  using  of  course  the  relation  that 

sin  i  =  n.  sin  r 

n  being  the  index  of  refraction  of  the  drop  for  the  particular 
wavelength  we  are  considering,  the  graph  will  come  out  to  be 
like  that  of  figure  86.  For  increasing  values  of  i,  the  value  of 
D  first  decreases,  then  increases.  If  we  take  n  =  1.33,  which 


RAINBOWS 


169 


61* 

Figure  86 


is  correct  for  a  certain  part  of  the  yellow,  the  minimum  value 
of  D  is  138°,  and  it  occurs  when  i  =  61°.  This  property,  that 
D  has  a  minimum  value,  is  of  0 
great  importance,  for  evidently  it 
rays  having  a  deviation  in  the  i 
immediate  neighborhood  of  138° 
would  be  far  more  numerous  than 
those  having  a  deviation  consid- 
erably more  than  this,  and  there 
are  none  at  all  with  a  deviation 
less  than  this.  Therefore,  al- 
though not  all  of  the  light  of  the  wavelength  in  question  is  sent 
out  from  the  drop  in  one  direction,  most  of  it  is  sent  out  very 
nearly  in  one  direction. 

Now  a  ray  having  a  deviation  138°  would  come  to  the 
observer's  eye  as  from  a  direction  making  an  angle  of  180°  — 
138°  =  42°  with  a  line  drawn  from  the  sun  through  the  eye, 
as  shown  in  figure  87.  Therefore  all  drops  situated  upon  a 
cone  of  half -angle  42°,  with  the  'eye  as  apex,  would  send  to  the 

eye  a  greater  amount  of 
light  of  this  wavelength 
than  any  other  drops,  and 
the  observer  would  see  a 
yellow  circular  arc.  If  we 
take  red  light,  whose  in- 
dex of  refraction  is  small- 
er, the  angle  of  minimum 
deviation  would  be  smaller 
than  138°,  and  therefore 
the  half  angle  of  the  cone 
on  which  lie  the  drops 
giving  maximum  red  light  would  be  greater  than  42°,  that  is, 
the  red  circle  would  be  outside  the  yellow,  and  correspondingly 
the  circles  for  shorter  wavelengths  would  lie  inside.  Therefore 
the  bow  is  red  on  the  outside  and  violet  on  the  inside.  Notice 
that  the  drops  giving  the  maximum  red  light  are  not  the  same 
as  those  giving  the  maximum  of  other  colors,  and  that  if  the 
observer  changes  his  position  the  drops  which  formerly  sent  to 
him  any  particular  color  will  no  longer  be  in  the  proper  direc- 
tion to  do  so,  and  their  function  will  be  supplied  by  other 


138'-' 


Figure  87 


170  LIGHT 

drops  which  are  in  proper  relation  to  his  new  position.  This 
accounts  for  the  bow  appearing  to  be  at  an  infinite  distance. 
Notice  also  that,  because  a  drop  sends  riot  all,  but  only  the 
greater  part,  of  its  light  in  one  general  direction  for  a  given 
wavelength,  there  will  be  a  great  deal  of  overlapping  of  colors. 
A  given  color  overlaps  all  the  other  colors  that  lie  within  it, 
that  is,  red  overlaps  all  the  others,  orange  overlaps  yellow,  green, 
blue  and  violet,  yellow  overlaps  green,  blue  and  violet,  etc. 
Only  the  red  is  even  approximately  pure. 

The  angular  radius  of  the  bow  is  somewhat  modified  from 
the  figures  given  above  (42°  for  the  yellow,  etc.)  by  the  inter- 
ference phenomena  already  mentioned,  which  we  shall  not  dis- 
cuss here,  and  by  the  fact  that  the  sun  is  not  a  point-source 
but  subtends  a  definite  angle. 

Not  all  the  light  that  enters  a  raindrop  traverses  such  a 
path  as  is  shown  in  figure  85.  Some  is  reflected  at  the  first 
incidence,  without  entering  the  drop.  Some  is  refracted  out 
where  it  strikes  the  surface  the  second  time,  without  any  internal 
reflections,  and  this  light  might  also  form  a  bow,  but  that  there 
is  for  this  case  nothing  like  a  minimum  deviation.  Some  light 
is  refracted  out  of  the  drop  after  two  internal  reflections,  some 
after  three,  etc...  and  theoretically  there  should  be  a  bow  for 
each  such  case.  The  second  bow,  that  formed  after  two  inter- 
nal reflections,  is  in  fact  often  seen.  It  is  formed  outside  the 
first  bow,  and  has  its  colors  in  reverse  order,  and  it  is  of  course 
fainter.  The  theory  of  its  formation  is  quite  similar  to  that 
for  the  first,  except  that  the  light  which  forms  it  comes  into 
the  drop  from  the  lower  side  and  emerges  from  the  upper  side. 
The  third  and  fourth  bows,  possible  in  theory,  are  still  fainter 
than  the  second,  and  to  make  matters  still  more  unfavorable 
they  come  in  the  bright  part  of  the  sky  near  the  sun.  There- 
fore they  cannot  be  seen.  Bows  of  still  higher  order  are 
inherently  too  faint  for  visibility. 

Rainbows  are  often  classed  with  other  natural  optical 
phenomena  of  the  atmosphere,  such  as  halos,  coronas,  mock- 
suns,  mirages,  etc.,  under  the  general  heading  of  "meteorologi- 
cal optics."  A  good  account  of  many  of  these  phenomena  is 
given  in  Wood's  "Physical  Optics,"  or  in  "W.  J.  Humphreys' 
"Physics  of  the  Air,"  part  III. 


MICHELSON  AND  MORLEY  EXPERIMENT        171 

84.  Motion  relative  to  the  ether. — In  section  68  it  was 
shown  how  the  Doppler  effect,  applied  to  the  spectrum  of  a 
star,  gives  us  a  means  of  finding  the  relative  velocity  of  the 
star  with  respect  to  the  earth.  It  was  also  stated  that  veloci- 
ties measured  in  this  way  are  usually  corrected  for  the  earth's 
orbital  velocity  so  as  to  give  the  star's  velocity  referred  to  the 
sun.  "Whether  the  sun  itself  is  in  motion  or  at  rest  is  a  ques- 
tion without  sense,  for  we  cannot  specify  motion  without  refer- 
ring to  some  body  which  we  regard  as  fixed.  In  other  words 
we  can  conceive  of  motion  only  as  a  relative  matter,  and  abso- 
lute motion  is  an  absurdity. 

But  it  is  not  absurd,  on  the  face  of  things,  to  speculate 
whether  the  sun  is  at  rest  or  in  motion  with  regard  to  the 
ether,  since  up  to  the  present  we  have  regarded  that  medium 
as  if  it  partook  in  some  measure  of  the  nature  of  ordinary 
material  things.  A  passenger  in  an  airplane  or  submarine 
could  readily  tell  that  he  was  moving  with  respect  to  the  sur- 
rounding air  or  water,  even  without  taking  note  of  surround- 
ing solid  objects,  and  it  is  conceivable  that  some  optical  ex- 
periment might  enable  us  to  detect  and  measure  our  velocity 
with  respect  to  the  ether. 

In  1887  Michelson  and  Morley  carried  out  an  experiment 
devised  with  the  above  purpose  in  view.     It  was  based  on  the 
assumption,   already  tacitly   made 
in  the  discussion   of   the    Doppler 
effect,   that  the  velocity  of  light- 
waves hi  the    ether,    like    that    of 
sound-waves  in  air,  does  not  par- 
take of  the  velocity  of  the  body 
which   emits  them.     If,   in  figure 
88,  a  body  moving  from  Q  toward 
P  emits  a  wave  when  it  is  at  the 
point  X,  this    wave    spreads    out 
into  an  enlarging  circle  whose  cen- 
ter remains  at  X  and  does  not  follow  the  emitting  body.    If  the 
circle,  with  X  as  center,  has  a  radius  equal  to  c,  the  velocity  of 
light,  then  it  would  represent  the  wavefront  one  second  after 
emission.   If  the  velocity  of  the  body  itself  is  v,  then  at  the  end 
of  the  second  it  would  be  at  the  point  Y,  where  XY  =  v.     If 


172 


LIGHT 


this  is  the  case,  then  to  a  person  moving  with  the  emitting 
body  the  velocity  of  light  should  appear  to  be  unequal  in 
different  directions.  Toward  P  the  velocity  would  be  YP  = 
c  —  v,  toward  Q  it  would  be  YQ  =  c  +  v,  and  toward  R,  at 
right  angles  to  the  line  QP,  it  would  be  YR  =  \/c2  —  v*".  Now 
refer  to  the  diagram  of  the  Michelson  interferometer,  figure  83, 

remembering  that  the  inter- 
ference is  between  two  beams 
going  from  the  diagonal  mir- 
ror EF,  one  up  to  AB  and 
back,  the  other  to  CD  and 
back.  Let  the  instrument  be 
adjusted  till  these  two  paths 
have  exactly  the  same  length, 
and  then  suppose  the  whole 
interferometer  to  be  mounted 
upon  some  base  which  is  mov- 
ing rapidly  to  the  right  in 
figure  83  with  a  velocity  v. 
The  two  paths,  in  spite  of  being  equal  in  length,  would  then 
not  include  the  same  number  of  wavelengths,  because  the  veloci- 
ty of  the  light,  with  respect  to  the  interferometer,  would  be 
different  in  different  directions.  From  the  diagonal  mirror  to 
AB  and  back,  it  would  be  Vc2  —  y2  5  from  the  diagonal  mirror 
to  CD  it  would  be  c  +  v ;  from  CD  back  to  the  diagonal  mirror 
it  would  be  c  —  v.  Consequently,  the  fringes  would  be  slightly 
shifted  from  their  position  when  the  instrument  was  at  rest 
with  respect  to  the  ether.  If  it  were  turned  through  90°,  so 
that  the  path  from  EF  to  AB  lay  along  the  direction  of  the 
velocity  v,  the  shift  would  be  in  the  opposite  direction.  Instead 
of  calculating  the  amount  of  the  shift,  we  shall  merely  calcu- 
late the  difference  in  the  time  for  the  two  interfering  beams 
of  light  to  come  together  again  after  the  separation.  If  d  is 
the  distance  from  the  diagonal  mirror  to  either  of  the  mirrors 
AB  and  CD,  the  time  required  for  the  beam  that  goes  up  to 
AB  and  back  is 


Figure    83 


2d 


RELATIVITY  THEORY  173 

approximately,  while  that  required  for  the  other  beam  is 

d  d  2cd          2d  v2       v4  2d   .      ,   v2 

+-v^c-^-?=V^^(1  +  ?-fc-  +  etC-)=T(1  +  ^ 

approximately.  The  terms  in  the  fourth  and  higher  powers  of 
v/c  have  been  dropped  because  any  attainable  velocity  of  matter 
is  so  small  compared  to  that  of  light. 

The  difference  in  these  two  times  is  v2d/c3.  If  we  use  for 
v  the  orbital  velocity  of  the  earth,  this  difference  is  exceedingly 
small.  But  the  interferometer  is  an  exceedingly  sensitive  in- 
strument, and  Michelson  and  Mbrley  were  able  to  increase  its 
sensitiveness  by  introducing  a  number  of  additional  reflections 
whose  effect  was  to  increase  the  distance  d.  The  whole  was 
then  mounted  upon  a  stone  slab  floated  in  mercury  so  that  it 
could  be  turned  about  a  vertical  axis.  It  was  turned  so  that 
first  one  arm,  then  the  other,  was  parallel  to  the  earth's  orbital 
motion,  but  the  expected  shift  of  the  interference  fringes  did 
not  take  place. 

85.  The  relativity  theory. — This  failure  of  Michelson  and 
Morley's  experiment  was  a  crisis  in  the  history  of  physics. 
There  can  be  no  doubt  that  the  earth  is  actually  in  motion, 
and  that  the  apparatus  should  have  detected  this  motion  if  no 
blunder  was  made  in  the  theory.  If  light  takes  on  the  velocity 
of  the  emitting  body,  as  it  would  if  Newton's  corpuscular 
theory  were  correct,  then  the  failure  of  the  experiment  is  what 
we  should  expect,  but  this  seems  impossible  if  light  consists  of 
waves.  Some  physicists  indeed  have  favored  throwing  over 
the  wave  theory,  and  the  ether  with  it,  but  this  cannot  be  done 
in  view  of  all  the  phenomena  of  interference.  Fitzgerald  and 
Lorentz  have  pointed  out  that  the  Michelson  and  Morley  ex- 
periment can  be  reconciled  with  the  wave  theory  if  we  assume 
that  ;a  body  in  motion  is  very  slightly  shortened  in  all  those 
dimensions  that  lie  parallel  to  its  motion.  Later  a  complete 
doctrine  known  as  the  " Relativity  Theory"  was  worked  out, 
based  on  two  fundamental  postulates:  first,  that  it  is  impossi- 
ble to  detect  by  any  means  any  relative  motion  between  matter 
and  the  ether, — second,  that  the  velocity  of  light  in  the  free 
ether  will  always  come  out  the  same,  no  matter  under  what 
circumstances  it  is  measured.  From  these  postulates  the  change 
of  dimensions  of  a  moving  body  suggested  by  Fitzgerald  and 


174  LIGHT 

Lorentz  follows  as  a  necessary  consequence,  but  it  also  follows 
that  the  mass  of  a  body  is  changed,  and  that  the  time-unit  is 
altered,  by  motion.  All  these  changes  are  exceedingly  small 
unless  the  velocity  becomes  comparable  with  the  velocity  of 
light. 

Einstein  has  in  recent  years  extended  the  relativity  prin 
ciple  to  accelerated,  as  well  as  to  steady  motion,  as  a  result  of 
which  he  was  led  to  predict,  among  other  things,  that  a  beam 
of  light  in  passing  near  a  body  with  a  strong  gravitational 
field,  like  the  sun,  would  be  deflected.  This  prediction  has 
been  apparently  verified  by  observations  during  a  recent  solar 
eclipse.  The  complete  theory  involves  fundamental  changes  in 
our  idea  of  time  and  space,  and  is  very  metaphysical  and 
mathematical,  as  well  as  physical. 

Problems. 

1.  Show  that,  in  the  circular  fringes  produced  by  uniform 
films,  a  ring  of  small  angular  diameter  corresponds  to  interfer- 
ence of  a  high  order,  that  is,  a  high  value  of  N.     Find  the 
highest    order   of   interference    for  a  film  of  glass,  index  1.54, 
1/10  mm.  thick,  for  light  of  A  .00005  cm. 

2.  What  would  be  the  effect  upon  the  fringes  seen  in  the 
Michelson  interferometer,  of  inserting  a  very  thin  slip  of  glass 
into  one  of  the  arms,  say  between  CD  and  EF  of  figure  83? 

3.  Show  that,  in  the  phenomenon  of  Newton's  rings,  the 
radii  of  successive  dark  rings  are  approximately  proportional 
to  the  square-roots  of  the  successive  integers,  assuming  that  the 
thickness  of  the  film  alone  determines  the  interference. 

4.  Prove  the  statement  in  paragraph  83,  that  D  =  180°  + 
2i  —  4r. 


CHAPTER  X. 

86.  Simple  harmonic  motion.— 87.  Velocity  in  S.  H.  M. — 88.  Accel- 
eration in  S.  H.  M. — 89.  Energy  in  S.  H.  M. — 90.  Two  parallel  S.  H. 
M.'s. — 91.  Application  to  cases  of  interference. — 92.  Two  S.  H.  M.'s  at 
right-angles. — 93.  Lissajous  figures. 

86.  Simple  harmonic  motion. — In  all  wave  phenomena,  we 
are  much  concerned  with  a  particular  kind  of  vibratory  motion 
known  a£  simple  harmonic  motion,  and  this  name  will  occur 
so  frequently  in  this  chapter  that  we  shall  at  the  outset  adopt 
for  it  the  abbreviation  S.  H.  M.  Its  definition  is  as  follows: 

The  motion  of  a  body  P,  figure  .89, 
is  simple  harmonic  along  a  line  MN  when 
it  moves  so  that  if  a  body  0  be  imagined 
travelling  at  a  uniform  rate  in  a  circle 
with  NM  as  diameter,  P  keeps  always  at 
the  foot  of  a  perpendicular  drawn  from 
O  to  MN.  The  time  in  which  0  com- 
pletes the  circuit,  which  is  the  same  as 
the  time  in  which  P  passes  from  M  to  N  Figure  89 

and  back  again,  is  called  the  period,  and  will  be  represented  by 
the  letter  r.  The  distance  CM,  half  the  range  of  motion  P,  called 
the  amplitude,  will  be  represented  by  K.  The  distance  CP,  repre- 
sented by  y,  from  the  middle  position  of  P  to  that  position 
which  it  momentarily  occupies,  is  the  displacement;  and  the 
corresponding  angle  OCM,  represented  by  <j>,  is  the  phase.  K 
and  T  are  constants  for  the  motion,  while  y  and  <£  are  variable 
with  the  time.  We  consider  y  to  be  positive  when  P  is  above 
C.  negative  when  it  is  below,  so  that  y  varies  from  K  to  — K, 
oscillating  between  these  values. 

Our  first  problem  will  be  to  find  a  mathematical  relation 
between  the  displacement  y  and  the  time,  represented  by  t. 
Since  P  is  at  the  foot  of  the  perpendicular  from  0,  cos.  <£  — 
CP/CO,  therefore 

y  =1  K.  cos  (j> 

Since  0  moves  uniformly  in  a  circle,  the  time  required  for  it 
to  move  through  the  angle  <£  will  be  to  the  time  necessary  to 

(175) 


176  LIGHT 

complete  the  circuit,  as  <f>  is  to  2?r.  Therefore,  if  we  are  to 
count  time  from  the  instant  when  P  is  at  M,  the  end  of  its 
path,  t  :  T  :  :  (j>  :2ir,  or 

4  —  27rt/T 

and 

IT        2?rt 
y  =  K.  cos  — 

Often,  however,  it  is  convenient  to  count  time,  not  from  a 
particular  instant  such  as  that  when  P  is  at  the  end  or  at  the 
middle  of  its  path,  but  from  some  instant  chosen  at  random, 
as  when  P  is  at  such  a  point'  as  Q  and  the  corresponding  posi- 
tion for  0  is  at  S.  Our  formula  may  easily  be  modified  to  suit 
this  more  general  condition.  Represent  the  angle  SCM  by  ft. 
Then,  since,  during  the  time  t,  0  has  moved  through  the  angle 
SCO  =  <£  +  £,  instead  of 

t  :  T  :  :  <£  :  2,r 
we  must  write 

t  :r  :  :  (<f>  + /3]   :  2,r 
giving 


and 

—  K  cos^—  —  B\  (1) 

\  a         .  7 

This  is  the  required  mathematical  relation  between  y  and 
the  time,  and  may  be  called  the  equation  of  the  S.  H.  M.  y 
could  easily  enough  have  been  put  into  the  form  of  a  sine 
instead  of  a  cosine,  by  simply  considering  the  complement  of 
<f>  instead  of  <f>  itself.  We  might  then  have  simply  defined  a 
S.  H.  'M.  at  the  start  as  a  motion  in  which  the  displacement 
is  a  sine  or  cosine  function  of  the  time,  and  so  avoided  speak- 
ing of  the  body  0  of  the  figure,  which  is  purely  imaginary, 
and  was  introduced  only  to  make  the  definition  easier  to  think 
of  physically. 

There  is  a  close  relation  between  equation  (1)  above  and 
equation  (5)  of  chapter  III,  which  is 

yrnK.COS—    (x  — Vt  — e) 


SIMPLE  HAHMONIC  MOTION  177 

The  latter,  since  it  is  the  complete  equation  for  a  wave,  in- 
volves, besides  the  variables  y  and  t,  the  third  variable  x, 
which  is  the  distance  from  a  fixed  origin,  measured  in  the 
direction  in  which  the  wave  is  progressing.  But,  if  we  fix  our 
attention  upon  a  definite  position  in  the  medium,  and  consider 
only  the  motion  there,  x  will  be  regarded  as,  for  the  time 
being,  a  constant,  and  therefore  we  may  bracket  it  with  the 
other  constant  e.  The  equation  then  becomes 

r2,rVt       2*,  J 

y  =  K.  cos     ---  (x  —  e) 
L     a  a  J 

(The  sign  of  the  quantity  whose  cosine  is  being  taken  has  been 
changed,  because  the  cosine  of  a  positive  angle  and  the  cosine 
of  an  equal  negative  angle  are  always  the  same.)  Now  V  is 
the  velocity  of  the  wave  and  a  is  the  wavelength,  as  can  be 
seen  by  referring  back  to  section  21,  and  therefore  a  =  VT  or 

V/a=l/r 
Therefore  we  can  write 


[2irt       2*  ,  1 

y  —  K  cos    ---  (x  —  e) 
Ira  J 


This  is  exactly  the  same  form  as  equation  (1)  above,  the  place 
of  ft  being  taken  by  2?r(x  —  «)/a.  Therefore  any  definite  part 
of  the  medium  must,  when  a  monochromatic  wave  passes 
through  it,  go  through  a  S.  H.  M.  The  value  of  the  phase- 
constant  ft  is  greater  the  farther  along  be  the  point  considered ; 
that  is,  although  a  point  far  from  the  source  goes  through  the 
same  motion  as  a  point  nearer,  it  is  behind  it  in  phase. 

The  relation  between  y  and 
t,  expressed  in  equation  (1),  is 
very  well  shown  in  a  graph  such 
as  the  heavy  curve  D  in  figure 

Si-r   ^      *      ^    \   ^         90,  where  t  is  the  abscissa  and  y 

the  ordinate.    It  is  the  same  kind 
\    /  of  curve  as  those  shown  in  figure 

19,  except  that  there  the  abscissa 
was  distance,  while  here  it  is  time. 

Evidently  the  maximum  height  or  depth  of  the  curve,  measured 
from  the  horizontal  axis,  is  the  amplitude  K,  while  the  distance 


178  LIGHT 

ac  is  the  period  T.  Those  portions  of  the  curve  that  slope  up- 
ward to  the  right  indicate  that  P  in  figure  89  is  moving  up, 
while  those  that  slope  downward  to  the  right  indicate  that  P  is 
moving  downward.  The  graph  is  drawn  not  only  to  the  right  of 
the  origin,  but  also  to  the  left,  corresponding  to  negative  times, 
for  we  may  certainly  suppose  that  the  motion  was  going  on 
before  the  instant  from  which  we  measure  time.  In  fact  fO 
is  the  time  it  took,  or  would  take,  for  P  to  move  from  C  out 
to  Q,  and  Om  is  'the  time  to  move  from  Q  to  the  end  of  ttie 
path. 

87.  Velocity  in  S.  H.  M. — The  simplest  considerations  show 
that  the  velocity  of  the  moving  point  P  must  be  zero  at  either 
end  of  its  path,  and  greatest  when  the  displacement  y  is  zero. 
Since  P  is  moving  sometimes  upward  and  sometimes  downward, 
we  shall  agree  to  call  the  velocity,  v,  positive  when  P  is  moving 
upward,  whether  it  is  actually  above  C  or  below  it,  and  nega- 
tive in  the  contrary  case.  Then  if  we  draw  another  graph,  in 
which  the  abscissa  again  represents  time  but  the  ordinate  the 
corresponding  velocity,  it  will  be  similar  to  the  displacement' 
curve,  having  the  same  period,  but  it  will  cross  the  axis  at 
points  where  the  displacement  is  greatest,  in  a  positive  or  a 
negative  direction. 

The  velocity-curve  could  be  constructed  graphically  from 
the  displacement-curve  by  means  of  the  following  considera- 
tions: A  velocity  is  the  ratio  of  a  very  short  distance  to  the 
very  short  time  in  which  that  distance  is  traversed,  that  is 

v  =  limit  <^ 
At 

The  symbol  /j  placed  before  a  quantity  means  a  very  small 
increase  in  that  quantity,  so  that  zlt  indicates  the  time 
in  which  the  displacement  of  P  increases  by  the  small  amount 
Ay-  But  in  a  graph  with  Cartesian  coordinates,  the  slope  of 
the  tangent  is  also  the  limit  of  Ay /At-  Therefore,  the  velocity 
of  P  at  any  instant  t  can  be  gotten  from  the  displacement- 
curve  by  simply  measuring  the  slope  of  the  tangent,  at  a 
horizontal  distance  t  from  the  origin.  If  the  slope  found  be 
erected  as  another  ordinate,  the  curve  so  constructed  will  be 
the  required  velocity-curve.  The  curve  V  in  figure  90  is  con- 
structed in  this  way.  The  amplitude  is  not  necessarily  the 


VELOCITY  CURVE  179 

same  as  for  the  displacement-curve,  and  whether  it  is  greater 
or  less  depends  upon  the  value  of  the  period.  It  is  evident 
from  figure  90  that  a  shortening  of  the  period  would  increase 
the  slope  of  the  D-curve,  and  also  from  physical  considerations 
it  is  clear  that,  the  shorter  the  period,  the  greater  would  be 
the  velocity  with  which  the  body  swings  through  its  middle 
position. 

The  mathematical  formula  for  the  velocity-curve  is  derived 
as  follows:  Suppose  the  time,  initially  represented  by  t,  in- 
creases by  a  small  amount  At.  The  initial  displacement  is 


The  displacement  after  the  change  in  time  is 

/    27Tt  ,      27TZlt    \ 

y  +  /ly  =  K.  cos  (  —  —  p  +  —  —  J 

The  last  equation  may  be  expanded  by  making  use  of  the  trigo- 
nometrical relation  for  the  cosine  of  the  sum  of  two  angles, 
cos  (x  -j-  y)   =  cos  x  cos  y  —  sin  x  sin  y 

letting       --  ft      take  the  place  of  x  and    '  that  of  y. 

We  then  have 


K.  sin 


/2irt  \    .       ^At 

I  ——P  jsin  —  ^- 


Since  we  are  to  find  the  limit  of  the  ratio  Ay  /At,  in  which 
At  approaches  zero,  and  since  the  cosine  of  an  angle  approaches 
unity  and  the  sine  approaches  the  value  of  the  angle  itself 
when  the  angle  becomes  very  small,  we  may  make  the  follow- 
ing substitutions: 


cos  -  =  sin 

T  T 


Therefore,  in  the  limit, 

/  2rrt          _  \  27rzlt  /  2irt  \ 

y  +  Jy  =  K.cos(—  —  ft  J  --  —  .  K.sm^—  —  ft  J. 
If  we  subtract  from    this    equation    the    equation    giving    the 


value  of  y  itself,  we  get 


180  LIGHT 


_      .     /2,rt 

Ay  —  --  .  K.sml   -- 

r  \    r 

and  if  we  divide  by  Jt,  we  get 

Ay  2-rrK     ,      /2irt  \ 

v  —  limit  —7  =  --  sin     --  8]  (2) 

At  T  \    r  / 

This  is  the  equation  for  the  velocity-curve  V  in  figure  90,  or 
the  formula  for  the  velocity. 

88.  Acceleration  in  S,  H.  M.  —  It  is  necessary  also  to  con- 
sider the  acceleration,  or  rate  of  change  in  velocity,  of  a  body 
in  S.  H.  M.  It  is  defined  as 

a  =  limit  — 

At 

therefore  it  bears  the  same  relation  to  the  velocity  as  the  latter 
bears  to  the  displacement,  and  the  acceleration-curve  can  be 
drawn  by  plotting,  for  each  value  of  the  time,  the  correspond- 
ing value  of  the  slope  of  the  V-curve.  The  dotted  curve  A  is 
obtained  in  this  manner.  From  what  has  already  been  said 
of  the  relation  between  the  V-curve  and  the  D-curve,  it  follows 
that  the  A-curve  will  have  its  peak  just  where  the  V-curve  is 
crossing  the  axis  on  the  rise,  and  this  of  course  entails  that  the 
peak  of  the  A-curve  and  the  trough  of  the  D-curve  come  at  the 
same  time,  —  in  other  words,  the  acceleration  and  the  displace- 
ment are  exactly  opposite  in  phase.* 

The  equation  for  the  A-curve  can  be  derived  quite  easily, 
by  the  same  method  we  used  for  the  V-curve.  The  velocity  v 
at  a  given  instant  t  -is 

*It  is  sometimes  difficult  for  a  student  to  conceive  that  the  body  P, 
figure  89,  actually  has  its  greatest  acceleration  when  it  is  at  the  bottom 
of  its  path,  and  the  velocity  is  momentarily  zero.  It  must  be  borne  in 
mind  that  just  before  P  reaches  the  bottom  its  velocity  is  downward 
in  direction,  while  just  afterward  it  is  upward,  so  that  it  is  precisely 
at  this  point  that  the  velocity  is  changing  most  rapidly.  There  is  no- 
thing incompatible  in  a  body  having  a  considerable  acceleration  when 
its  velocity  (for  a  single  instant  only)  is  zero.  For  instance,  a  stone 
thrown  into  the  air  has  the  acceleration  980  cm.  per  sec2,  even  at  the 
instant  when  it  is  at  the  top  of  its  path. 


ACCELERATION  CURVE  181 


2^K          /2rf  \ 

=  ---  .  sm     —  —  R 

\    r  '     / 


let  t  be  increased  by  the  very  small  amount  At,  and  let  Jv  be 
the  corresponding  increase  in  v.     The  new  value  of  v  is 


L    r  r     J 

Remembering  that  sin  (x  +  y)  =  sin  x.  cos  y  +  cos  x.  sin  y,  and 

substituting   '     -  —  (3     f or  x  and   — for  y,  we  have 

2irK  /27rt          _  \  27rjt       2?rK  /27rt  \  2irZ 

f-  Av  = .  sm  (  —  —  p  ).  cos .  cos  ( B  ) .  sm  — 

T  \     T  /  T  T  \    *  /  T 

Again  considering  that  we  are  to  take  the  case  of  the  limit, 
we  substitute 

27rZlt                        27rZlt       2*  At 
cos  —  1  sin =  

T  T  T 

27rK        .       /  2irt  \        47r2K/(t  /  2^rt 

T  \      T  /  T2  \       T 

Subtracting  the  value  of  v  itself, 

4rr2Zlt  /27rt 

T2  \      T 

and  dividing  by  Jt, 

a  =  limit  ——  = —  cos 

At  r2 

Another  form  for  the  equation  may  be  obtained  by  sub- 
stituting y  instead  of  the  factor 


which  gives 

47T2 

a^-y  (4) 


182  LIGHT 

The  minus  sign  in  equation  (4)  indicates  that  whenever 
the  displacement  is  upward  the  acceleration  is  downward,  and 
vice  versa,  which  means  of  course  that  the  acceleration,  and 
therefore  the  force  acting,  are  always  such  as  to  drive  the  body 
back  to  its  central  position,  C.  The  equation  also  shows  that 
the  acceleration  is  directly  proportional  to  the  displacement, 
the  factor  of  proportionality  being  4-n-2  divided  by  the  square 
of  the  period. 

89.  Energy  in  S.  H.  M.—  A  body  in  S.  H.  M.  is  of  course 
.endowed  with  a  certain  amount  of  energy,  which  will  be  con- 
stant unless  there  are  some  disturbances.  At  the  end  of  the 
swing  the  energy  is  all  potential,  while  at  the  middle  it  is  all 
kinetic,  and  at  other  positions  it  is  partly  potential  and  partly 
kinetic.  The  simplest  way  to  find  an  expression  for  the  total 
energy  would  be  to  find  the  potential  energy  at  the  end  of  the 
swing  or  the  kinetic  at  the  middle.  We  shall  do  both,  and 
show  that  the  same  expression  is  found  in  each  case. 

The  potential  energy  at  the  end  of  the  path  is  the  work 
that  must  be  done  to  pull  the  body  out  to  that  point,  starting 
with  it  at  rest  in  the  central  position.  Work  is  force  applied 
times  the  distance  moved,  and  the  force  is  the  body's  mass,  m, 
multiplied  by  the  acceleration  which  the  force,  acting  alone, 
would  produce  in  it.  If  the  body  be  pulled  out  slowly  enough, 
this  acceleration  will  be  just  equal  and  opposite  to  that  which 
the  body  has  when  freely  swinging,  which  we  have  found  to  be 
—  47r2y/T2.  Therefore  the  force  applied  must  be  the  mass  m 
multiplied  by  the  equal  and  opposite  acceleration  -)-  47r2y/r2. 
That  is 


F  rr: 


We  cannot  get  the  work  by  mul- 
tiplying this  expression  by  the 
whole  distance  moved,  K,  for  y  is 
a  variable,  and  therefore  F  is  also, 
having  the  value  zero  at  the  be- 
ginning of  the  motion  and  the 
value  of  47T2mK/r2  at  the  end.  We  must  calculate  the  work 
by  infinitesimal  steps,  F  having  a  different  value  for  each 
step.  To  do  this,  consider  figure  91,  in  which  OQ  is  a  graph 


ENERGY  OP  S.  H.  M.  183 

whose  abscissa  is  y  and  ordinate  the  corresponding  value  of  F. 
In  accordance  with  the  above  equation,  it  is  a  straight  line. 
The  distance  OP  represents  the  whole  distance  moved,  or  K, 
while  PQ  is  the  force  necessary  to  hold  the  body  still  at  the 
end  of  its  path,  ^rrmK/T2.  We  shall  prove  that  the  work  done 
in  pulling  the  body  out  to  the  end  of  its  path  is  equal  to  the 
area  of  the  triangle  OPQ. 

Recalling  that  the  work  must  be  calculated  step  by  step, 
let  us  see  how  much  work  is  done  in  moving  the  body  from  C 
to  C',  the  distance  CC'  being  supposed  very  small.  At  the 
beginning  of  this  step,  the  force  applied  is  CD,  and  if  it  con- 
tinued to  have  this  value  through  the  step,  the  work  would 
be  CD  X  CC',  which  is  the  area  of  the  rectangle  CDd'C'.  At 
the  end  of  the  step,  the  applied  force  is  C'D',  and  if  it  had 
this  value  throughout  the  step  the  work  would  be  C'D'  X  CC', 
the  area  of  the  rectangle  CdD'C'.  Evidently  the  actual  work 
done  during  the  small  motion  fronTC  to  C'  lies  between  these 
two  values,  and  when  we  consider  the  work  done  during  the 
whole  motion  from  O  out  to  P,  it  follows  that  this  lies  in  value 
between  the  area  of  the  stair-shaped  figure  extending  above 
the  line  OQ  and  that  of  the  similar  figure  all  of  which  is  below 
OQ.  If  the  steps  are  made  smaller  and  smaller,  the  area  of 
each  of  these  figures  comes  nearer  and  nearer  to  the  area  of  the 
triangle  OPQ,  therefore  we  can  say  that,  in  the  limit,  the  work 
done  is  equal  to  the  area  of  that  triangle,  that  is,  the  potential 
energy  at  the  end  of  the  swing,  or  the  total  energy  at  any  time, 


The  calculation  of  the  kinetic  energy  when  the  body  is 
passing  through  its  central  position  is  easy.  For  kinetic  energy 
is  one  half  the  mass  times  the  square  of  the  velocity,  and  the 
velocity  in  the  central  position  is  simply  the  greatest  value 
that  the  body  can  have,  which  is  gotten  from  equation  (2)  by 
putting  the  sine  equal  to  1.  Therefore  the  kinetic  energy  at 
this  point,  or  the  total  energy  of  the  body  at  any  time,  is 


the  same  expression. 


184  LIGHT 

The  fact  that  the  energy  of  a  body  in  S.  H.  M.  is  propor- 
tional to  the  square  of  the  amplitude  of  vibration  is  quite  im- 
portant, but  it  is,  after  all,  what  we  might  have  expected.  For, 
in  order  to  set  a  body  in  vibration,  first  with  an  amplitude  of 
1  inch,  and  later  with  an  amplitude  of  2  inches,  we  must  not 
only  pull  it  out  twice  as  far  in  the  second  case,  but  also  exert 
a  force  whose  average  value  is  twice  as  great  so  that  the  work 
done  is  four  times  as  great. 

90.  Two  parallel  S.  H.  M.'s. — In  examples  of  interference, 
and  in  other  problems  in  light  or  sound,  it  is  often  necessary 
to  consider  the  application  simultaneously  of  two  S.  H.  M.'s 
to  the  same  body.  We  shall  suppose  that  the  two  motions  have 
the  same  period,  and  are  in  the  same  direction,  but  they  may 
differ  to  any  degree  in  amplitude,  and  they  may  have  the  same 
phase,  opposite  phases,  or  any  desired  difference  in  phase.  An 
excellent  example  of  the  superposition  of  two  S.  H.  M.'s  in 
this  way,  where  the  motion  however  is  applied  to  an  illumi- 
nated spot  on  a  screen  instead  of  to  a  material  body,  is  shown 
in  figure  92.  H  is  a  hole  through  which  streams  a  beam  of 

L 


Figure  92 

light  from  the  sun  or  a  bright  artificial  source,  L  a  lens,  A 
and  B  two  tuning  forks,  S  a  white  screen.  The  forks  have 
the  same  period  and  are  arranged  to  vibrate  in  the  same  plane. 
Each  has  a  small  mirror  (Mt  and  M2)  mounted  on  one  prong, 
and  the  parts  are  arranged  so  that  the  light  is  reflected  from 
Mt  to  M2,  and  thence  to  the  screen,  where  it  comes  to  a  focus, 
forming  a  small  round  image  of  the  hole  H.  If  either  fork  is 
set  in  vibration  while  the  other  is  held  still,  the  illuminated 
spot  will  be  set  into  S.  H.  M.  If  both  forks  are  vibrating  to- 
gether, the  spot  will  also  have  a  vibratory  motion.  We  are  to 
find  out  whether  this  motion  is  simple  harmonic,  and  if  so 
what  its  amplitude  and  phase-constant  will  be. 

Suppose  that  the  motion  of  the  spot  due  to  the  first  fork 
alone  is  represented  by  the  equation 


PARALLEL  S.  H.  M.'S 


185 


=  K!  COS 

r 


that  due  to  the  second  fork  alone  by 


It  is  unnecessary  to  put  a  phase  constant  into  each  expression, 
since  we  may  suppose  that  the  zero-point  for  time  measurement 
is  so  chosen  that  the  phase-constant  of  the  first  motion  is  zero. 
Then  ft  is  simply  the  difference  in  phase,  or  the  amount  by 
which  the  second  fork  lags  behind  the  first  in  phase. 

The  resultant  motion,  when  both  forks  are  vibrating,  is 
gotten  by  adding  yt  and  y2,  for  evidently  the  movement  due 
to  one  fork  will  be  superimposed  upon  that  due  to  the  other. 
We  shall  discuss  the  problem  graphically,  making  use  of  the 
definition  of  S.  H.  M.,  that  it  is  the  projection,  upon  a  straight 
line,  of  uniform  motion  in  a  circle.  In  figure  93,  let  ba  be  the 
path  of  the  vibration 
due  to  the  first  com- 
ponent motion.  The 
large  circle,  of  which 
the  diameter  is  ab  and 
the  radius  equal  to  K1? 
will  be  called  the  ref- 
erence-circle for  the 
first  motion,  and  since 
we  have  taken  the 
phase  constant  for  that 
motion  as  zero,  it  alone 
would  for  time  zero  Figure  93 

put  the  spot  of  light  at  a,  the  end  of  the  diameter,  and  the 
tracing  point  moving  in  the  reference  circle  would  also  be  there. 
That  is,  yx  =  Kl  =  oa,  at  that  instant.  In  order  to  add  in  y2, 
the  component  due  to  the  second  vibration,  we  draw  a  reference - 
circle  with  a  as  center  and  K^  as  radius,  and  lay  off  the  angle 
eac,  equal  to  the  phase-difference  /?.  Then,  for  time  zero,  the 
second  component  of  the  motion  gives  a  displacement  y2  =  ad, 
the  projection  of  ac,  and  the  total  displacement  is  yl  4-  y2  =  od. 
Notice  also  that  od  is  the  projection  of  the  single  line  oc. 


186  LIGHT 

Now  consider  the  state  of  affairs  at  a  later  instant,  when 
the  tracing  point  in  the  first  reference-circle  has  moved  to  a', 
through  the  angle  aoa'.  Let  us  suppose  that,  along  with  the 
radius  oa,  the  whole  triangle  oac  rotates  about  the  point  o  as 
if  it  were  rigid,  taking  the  new /position  oa'c'.  Whenever  a 
rigid  plane  figure  rotates  about  a  point  in  its  own  plane,  all 
lines  of  the  figure  turn  through  the  same  angle  in  the  same 
time.  Therefore  the  angle  coc',  turned  through  by  the  side  oc, 
is  equal  to  the  angle  aoa',  turned  through  by  the  side  oa.  Also, 
if  c'a'  and  ca  be  produced  till  they  meet,  the  angle  so  formed 
will  also  be  equal  to  aoa'.  (These  statements  can  easily  be 
proved  by  simple  geometry.)  Since  the  two  S.  H.  M.'s  have 
the  same  period,  the  radii  oa  and  ac  must  necessarily  turn 
through  the  same  angle  in  the  same  time.  Therefore  a'c'  will 
be  the  direction  of  the  tracing-point  radius  in  the  second  motion 
at  the  same-  instant  when  oa'  is  that  for  the  first.  For  that 
instant,  then,  yl  must  be  equal  to  oh,  the  projection  of  oa',  y2 
to  hd',  the  projection  of  a'c',  and  the  total  displacement  is 

yx  +  y2  =?  oh  +  hd'  =  od' 

and  od'  is  nothing  but  the  projection  of  the  third  side  of  the 
triangle  oc',  in  its  new  position. 

The  same  result  holds,  whatever  may  be  the  angle  through 
which  oa  has  turned.  In  each  case,  the  resulting  displacement 
is  the  projection  of  the  third  side  oc  in  its  new  position.  Since 
this  is  the  projection  of  a  line  of  fixed  length,  rotating  with 
the  same  period  as  the  two  component  S.  H.  M.'s,  the  result- 
ing motion  is  itself  evidently  simple  harmonic,  of  the  same 
period  as  yt  and  y2,  of  amplitude  oc,  and  behind  the  first 
motion  yx  in  phase  by  the  angle  aoc,  or  <f>. 

"We  have  proved  that  the  following  rule  holds  for  finding 
the  resulting  motion  when  two  S.  H.  M.'s  of  the  same  period 
and  direction  act  upon  the  same  body:  Lay  off  a  line  oa  whose 
length  is  the  amplitude  of  the  first  motion.  From  a  lay  off 
another  line  ac,  whose  length  is  equal  to  the  second  amplitude, 
so  that  the  angle  it  makes  with  oa  produced  is  equal  to  the 
amount  by  which  the  second  component  lags  in  phase  behind 
the  first.  Then,  completing  the  triangle,  the  side  oc  gives  the 
resulting  amplitude,  and  the  motion  lags  behind  the  first  com- 
ponent in  phase  by  the  angle  aoc.  Since  this  is  exactly  the 


APPLICATION  TO  INTERFERENCE  187 

rule  for  finding  the  resultant  of  two  vectors,  the  angle  between 
them  being  the  phase-difference,  it  may  be  said  that  S.  H.  M.'s 
are  compounded  as  vectors,  but  it  should  be  remembered  that 
this  statement  refers  only  to  a  diagrammatic  representation, 
and  has  nothing  to  do  with  the  actual  directions  in  which  the 
vibrations  take  place. 

The  resultant  vibration  can  be  represented  by  the  formula 


y  =  K.  cos 


(¥-«) 


where  K  =  oc,  and  <£  =  angle  aoc.     Since,  from  trigonometry. 

oc2  =  oa2  -f-  ac2  -|-  2  X  oa  X  ac  X  cos(eac) 
it  follows  that 

K2  =  K,2  +  K22  +  2  K,  K2  cos  p        (6) 

If  both  the  component  motions  have  the  same  amplitude,  so 
that  K2  =  K, 

K2=r2K12  (l  +  cos0)  (7) 

If  p  =  0,  cos  p  =  1,  and  K2  =  tKf,  or  K  =  2K1?  as  can  be 
seen  either  from  equation  (7)  or  from  consideration  of  the 
above-mentioned  rule  for  finding  the  resultant  amplitude  by  a 
vector  diagram.  In  this  case  the  energy  of  the  resultant  S. 
H.  M.  is  4  times  that  of  either  component,  because  the  energy 
depends  upon  the  square  of  the  amplitude.*  If  ft  =  TT,  or  180°, 
then  cos/3  —  --1,  and  K  =  0,  as  shown  by  equation  (7)  or 
by  the  vector  diagram.  For  other  values  of  p,  K  will  lie  be- 
tween 0  and  2Kt. 

91.  Application  to  cases  of  interference. — These  results 
can  be  easily  applied  to  any  case  of  interference  where  only 
two  beams  of  light  are  concerned.  Take,  for  example,  the 
interference  with  Fresnel's  mirrors,  and  refer  to  figure  20. 
The  two  slits  St  and  S2  being  nearly  equally  distant  from  any 
portion  of  the  screen  AB,  we  can  regard  the  amplitude  of  the 
two  beams  as  being  equal.  At  C,  where  the  difference  in  path 

*In  the  arrangement  depicted  in  figure  92  there  is  no  energy  as- 
sociated with  the  movement  of  the  illuminated  spot  on  the  screen, 
since  this  motion  is  not  a  motion  of  a  real  thing,  but  only  a  trans- 
ferrence  of  illumination  from  one  place  to  another.  The  light  vibra- 
tions producing  the  illumination  of  course  have  energy,  and  so  do  the 
two  forks. 


188  LIGHT 

is  zero,  at  M±  or  M/  where  it  is  a  wavelength,  at  M2  or  M/ 
where  it  is  two  wavelengths,  etc.,  the  phase-difference  is  zero 
or,  what  amounts  to  the  same  thing,  an  integral  multiple  of 
2^r  (360°),  the  resulting  amplitude  is  twice  that  which  one 
beam  alone  would  produce,  and  the  brightness  four  times  as 
much.  At  the  points  mx  and  m/,  where  the  path  difference  is 
A/2,  at  m2  and  m./  where  it  is  3A/2,  etc.,  the  phase-difference 
is  either  TT  (180°)  or,  what  amounts  to  the  same  thing,  an  odd 
multiple  of  TT,  and  the  cosine  is  —  1.  These  points  therefore 
have  zero  resulting  amplitude,  and  they  will  be  quite  dark. 
At  other  points,  the  cosine  of  the  phase-difference  is  neither 
-j-  1  nor  — 1,  but  some  intermediate  value,  and  therefore  the 
brightness  will  be  neither  so  great  as  at  C  nor  absolutely  zero 
as  at  nij. 

The  vector  diagram  construction  for  finding  the  resultant 
of  two  S.  H.  M.'s  can  be  extended  to  cases  of  three  or  any 
number  of  component  motions,  provided  all  have  the  same 
period  and  all  are  in  the  same  direction.  We  simply  regard 
each  amplitude  as  a  vector,  the  direction  of  which  on  the  dia- 
gram is  given  by  the  phase-constant,  and  find  the  resultant,  the 
length  of  which  gives  the  amplitude  of  the  resultant  vibration. 
Many  rather  complicated  phenomena  of  interference  or  diffrac- 
tion can  be  easily  treated  by  this  method. 

If  the  student  will  refer  back  to  figure  69,  chapter  VII,  he 
should  recall  that,  in  the  discussion  of  a  grating  with  9  open- 
ings, it  was  found  that  at  the  .point  I  the  rays  from  all  the 
different  openings  arrived  together  in  phase,  while  at  points 
x  and  x',  close  to  I,  the  rays  opposed  one  another  in  pairs  by 
coming  together  in  opposite  phases,  that  is  with  a  phase-differ- 
ence which  is  an  odd  multiple  of  TT,  leaving  only  the  ray  from 
a  single  opening  to  exert  its  full  effect.  Therefore  the  ampli- 
tude of  the  resulting  S.  H.  M.  which  produces  the  light  at  I 
is  nine  times  that  at  x  or  x',  and  the  energy  of  vibration,  to 
which  the  brightness  is  proportional,  is  81  times  as  great  at  I 
as  at  x  or  x'. 

A  very  interesting  application  of  the  principles  we  have 
taken  up  in  this  chapter,  is  in  connection  with  the  light  sent 
out  by  a  luminous  source  such  as  a  flame.  The  actual  centers 
from  which  the  light  is  emitted  are  millions  of  radiating  atoms 
or  molecules,  and  since  these  are  independent  of  one  another 


INTENSITY  FROM  MANY  SOURCES  189 

in  their  vibration,  we  must  assume  that  all  sorts  of  phase- 
differences  exist  between  them.  It  might  be  argued  that  since 
one  of  these  radiating  centers  is  as  likely  to  have  one  phase- 
constant  as  any  other,  therefore  in  the  whole  assemblage  they 
should  annul  one  another's  effects,  with  the  result  that  no  light 
would  be  emitted  by  the  flame  at  all.  Such  a  contention  would 
be  as  erroneous  as  the  statement,  sometimes  loosely  made,  that  in 
a  sufficiently  large  number  of  throws  of  a  coin,  it  would  show 
heads  just  as  often  as  tails.  Both  are  due  to  a  misunderstand- 
ing of  the  laws  of  probability.  It  is  true  that  in  a  great  num- 
ber of  throws  of  a  coin,  the  difference  between  the  number  of 
heads  and  the  number  of  tails  becomes  less  and  less  as  com- 
pared to  the  total  number  of  throws,  but  the  absolute  amount 
of  the  difference  tends  to  increase.  In  fact,  the  probable  ex- 
cess of  heads  over  tails  or  conversely  of  tails  over  heads  (theory 
cannot  predict  which  it  will  be)  is  directly  proportional  to  the 
square-root  of  the  number  of  throws.  Similarly,  theory  pre- 
dicts that  in  the  case  of  a  great  number  of  radiating  centers, 
with  phases  distributed  at  random,  the  probable  amplitude  of 
the  resulting  vibration  determining  the  illumination  at  any 
place  will  be  proportional  to  the  square-root  of  the  number  of 
radiating  centers.  Therefore  the  energy  and  the  brightness  of 
illumination  should  be  proportional  to  the  number  of  centers. 
A  flame  of  double  size  should  then  give  double  illumination, 
other  things  being  equal,  and  this  is  what  is  actually  found. 

If  we  were  able  to  control  the  phases  of  all  the  centers 
of  radiation  in  a  flame,  so  that  all  the  rays  arrived  at  the  eye 
in  the  same  phase,  the  brightness  would  be  proportional  to  the 
square  of  the  number  of  centers,  and  therefore  would  be  enor- 
mously increased.  Under  these  circumstances  a  sodium  flame 
which  is  actually  very  feeble  would  appear  many  times  brighter 
than  the  sun  and  would  be  absolutely  destructive. 

A  similar  case  occurs  in  sound.  Suppose  that  in  an  or- 
chestra we  have  a  large  number  of  instruments  of  the  same 
kind,  say  100,  all  striking  the  same  note  at  once.  If,  by  any 
chance,  the  sound  waves  from  all  these  instruments  reached 
the  ear  in  the  same  phase,  the  intensity  of  the  sound  heard, 
proportional  to  the  square  of  the  resulting  amplitude,  would 
be  10000  times  as  great  as  that  heard  when  only  one  instrument 


190  LIGHT 

is  being  sounded,  and  the  result  would  be  deafening.  The 
chance  for  this  to  occur,  however,  is  almost  vanishingly  small. 
In  general,  the  phases  are  distributed  at  random,  and  the 
sound  heard  is  proportional  to  the  number  of  instruments. 

92.  Two  S.  H.  M.'s  at  right-angles  —  Another  kind  of  com- 
bination of  two  S.  H.  M.'s,  the  importance  of  which  in  the 
study  of  light  will  be  shown  in  chapters  XII  and  XIII,  is  one 
in  which  the  vibrations  are  at  right-angles.  There  are  many 
examples  in  physics.  For  instance,  a  ball  hung  from  a  string 
whose  upper  end  is  clamped  can  be  set  swinging  as  a  pendulum 
in  either  an  east-  west  or  a  north-south  direction,  and  there  is 
no  reason  why  both  motions  should  not  go  on  together.  From 
simple  reasoning,  it  seems  probable  that  the  resulting  motion 
would  in  general  be  elliptical,  and  we  shall  show  by  analysis 
that  this  is  true.  Another  good  illustration  is  gotten  by  sup- 
posing one  of  the  tuning-forks  of  figure  92  to  be  turned  through 
90°  a.bout  a  horizontal  axis,  so  that  its  vibrations  are  in  a 
vertical  plane,  the  other  still  vibrating  in  a  horizontal  plane. 
The  spot  of  light  would  then  be  subject  to  a  vertical  and  a 
horizontal  S.  H.  M.,  and  would  in  general  describe  an  ellipse. 
We  shall  assume,  as  before,  that  the  periods  are  the  same. 
Let  the  horizontal  and  the  vertical  vibrations  be  represented 
respectively  by 

2?rt 
X  —  A.  COS  -  (8) 

T 


y  —  B.  cosf  —  —  /?  j  (9) 

The  latter  equation  can  be  rewritten  in  the  form 

y  =  B.  cos  ft.  cos  —  '  -f-  B  sin  ft.  sin  —          (10) 

To  find  the  equation  of  the  path,  we  must  eliminate,  between 
(8)  and  (9)  or  between  (8)  and  (10),  that  variable  which  has 
no  place  in  the  simple  equation  of  a  line,  viz.,  the  time.  This 
can  best  be  done  by  deriving  from  (8) 

27rt         X 

cos  —  —  —      and      sin 
r  ^k 

and  substituting  these  values  in  (10).     This  gives 


S.  H    M.'S  AT  RIGHT  ANGLES 


Bx 


191 


y T-  cos  ft  =  B  \  1  —  --   in 


By  squaring  both  sides,  we  get 
y2  —  ^2  cos  p  +          C0s 


2xy 
AB 


(11) 


Equation  (11),  represents  an  ellipse,  except  when  sin/?==: 
0,  that  is,  when  the  phase  difference  ft  is  either  0  or  ?r.  If 
ft  =  0,  cos  ft  =  1,  and  the  equation  reduces  to 


A2       AB  +  B*  ~~  ° 


or, 


the  equation  of  a  straight  line  whose  slope  is  B/A.     If  ft  =  IT. 
cos  ft  =  —  1  and  the  equation  reduces  to 


a  straight  line  whose  slope  is  —  B/A. 


Figure  94 

Figure  94  shows  a  number  of  typical  shapes  that  the  path 
can  take  according  to  the  value  of  ft.    In  each  case,  the  whole 


192  LIGHT 

motion  must  lie  within  the  rectangle  of  the  dimensions  2A  X 
2B,  for  evidently  x  cannot  be  greater  than  -f-  A  nor  less  than 
—  A,  and  y  cannot  be  greater  than  +  B  nor  less  than  —  B. 
Except  in  the  cases  where  the  ellipse  becomes  a  straight  line, 
the  path  is  tangent  to  the  sides  of  the  rectangle.  Arrows  show 
the  direction  of  motion  in  the  path.  The  equation  of  the 
ellipse  cannot  tell  us  anything  about  the  direction  of  motion, 
for  the  time  has  been  eliminated,  but  it  can  be  determined  for 
each  case  by  reasoning  as  in  the  following  example:  "When 
/?  =  7T/2,  or  90°,  the  vertical  motion  must  be  %  period  behind 
the  horizontal,  that  is,  y  must  be  zero  but  increasing  when 
x  =  +  A.  A  quarter  period  later,  y  will  be  equal  to  +  B, 
and  x  =  0.  Since,  in  a  quarter  period,  the  point  moves  from 
the  right-hand  extreme  of  the  ellipse  to  the  upper  extreme,  the 
rotation  must  be  counterclockwise.  When  the  ellipse  reduces 
to  a  straight  line,  for  ft  equal  to  zero  or  TT,  the  motion  is  of 
course  back  and  forth  along  a  diagonal  of  the  rectangle. 

If  the  difference  in  phase  is  ir/2  or  37T/2,  and  if,  in  addition 
the  amplitudes  A  and  B  are  equal,  the  ellipse  becomes  a  circle, 
in  which  the  motion  is  counterclockwise  or  clockwise. 

As  a  converse  to  what  we  have  proved  about  the  produc- 
tion of  a  harmonic  elliptic  motion  as  a  result  of  the  superposi- 
tion of  two  linear  S.  H.  M.'s  at  right-angles,  it  may  evidently 
be  said  that  any  given  elliptic  harmonic  vibration  can  be  re- 
placed for  the  purpose  of  mathematical  analysis,  by  a  pair  of 
linear  S.  H.  M.  's  at  right-angles  which  would  produce  this  par- 
ticular elliptic  motion.  Consideration  of  figure  94  shows  that 
the  analysis  can  be  made  in  a  number  of  ways.  The  directions 
of  the  two  component  vibrations  that  are  to  replace  the  ellipse 
may  coincide  with  the  major  and  minor  elliptic  axes  or  be 
inclined  to  them  at  any  angle.  In  other  words,  there  are  many 
pairs  of  linear  motions  which  are  equivalent  to  a  given  ellip- 
tic motion,  though  the  amplitudes  and  phase-difference  are  of 
course  not  always  the  same.  In  most  cases,  we  wish  to  replace 
an  elliptic  motion  by  two  linear  motions  in  the  directions  of 
the  elliptic  axes,  and  then  the  amplitudes  will  be  equal  to  the 
semi-axes  of  the  ellipse,  and  the  phase-difference  will  be  7r/2  or 
37T/2  according  as  the  motion  is  counterclockwise  or  clockwise. 
It  is  also  plain,  from  the  cases  in  the  figure  for  which 


LISSAJOUS  FIGURES  193 

ft  z=  0  or  TT,  that  a  linear  S.  H.  M.  can  be  resolved  into  two 
other  mutually  perpendicular  linear  S.  H.  M.'s  either  in  the 
same  or  in  opposite  phases,  and  that  the  amplitudes  of  the 
components  is  gotten  by  exactly  the  same  rule  which  holds 
when  we  resolve  a  force,  or  any  other  vector,  into  two  mutually 
perpendicular  components.  For  such  a  resolution  then,  ampli- 
tudes of  S.  H.  M.'s  may  be  treated  as  vectors.  It  is  easy  to 
see  that  the  energy  of  the  original  vector  will  be  the  sum  of 
the  energies  of  the  two  components.  For,  since  the  amplitude 
of  the  former  forms  the  hypotenuse  of  a  right  triangle,  of 
which  the  component  amplitudes  are  sides,  the  square  of  the 
former  is  equal  to  the  sum  of  the  squares  of  the  latter,  and  we 
know  that  the  energy  is  proportional  to  the  square  of  the 
amplitude. 

93.  Lissajous  figures. — A  very  curious  and  beautiful  ap- 
pearance is  produced  when  two  linear  S.  H.  M.'s  at  right- 
angles,  combining  to  form  an  elliptic  motion,  have 
periods  which  are  nearly  the  same,  but  not  quite. 
The  difference  in  period  causes  one  to  gain  upon 
the  other  in  phase,  and  the  motion  passes  through 
all  the  forms  of  figure  94,  repeating  the  whole 
cycle  again  and  again.  The  figures  so  formed  are 
known  as  Lissajous  figures.  The  following  simple 
experiment,  which  anyone  can  perform,  shows  the 
cycle  of  changes  very  well  in  a  slowly  moving 
system.  A  ball  R,  figure  95,  is  hung  from  any  (!) 

support  by  cords  in  the  manner  shown.  It  is  then  Figure  95 
capable  of  swinging  as  a  pendulum  of  length  RQ  in  the  plane 
of  the  figure  and  as  one  of  length  RP  in  a  perpendicular  plane, 
and  the  two  motions  will  of  course  have  different  periods.  If  it 
be  started  swinging  in  a  direction  inclined  to  both  planes,  it 
will  slowly  pass  through  all  the  configurations  of  figure  94. 

Problems. 

1.  Calculate  the  energy  of  a  body  of  mass  1000  grams  in 
simple  harmonic  motion  of  period  %  sec.  and  amplitude  10  cm. 
\Yhat  is  the  force  acting  on  this  body  when  it  is  2  cm.  from  the 
center  of  its  path? 


194  LIGHT 

2.  Show  that  when  two  S.  H.  M.'s  in  th,e  same  straight 
line  are  applied  to  a  body,  each  having  the  same  amplitude  and 
period,  but  with  a  difference  in  phase  of  120°,  the  resultant 
motion  has  the  same  amplitude  as  either  component. 

3.  Show  that  the  maximum  kinetic  energy  of  a  body  in 
S.  H.  M.  is  equal  to  the  kinetic  energy  of  the  tracing  body 
(0  of  figure  89)  if  the  latter  has  the  same  mass. 

4.  ,Show  that  when  two  S.  H.  M.'s  at  right-angles,  with 
same  amplitude  and  phase-difference  of  ir/2,  combine  to  produce 
a  uniform  circular  motion,  the  energy  of  the  latter  is  equal  to 
twice  that  of  either  component. 

5.  Show  that  when  two  S.  H.  M.'s  at  right-angles,  with  a 
phase-difference  of  0  or  TT,  combine  to  produce  a  linear  S.  H. 
M.  in  another  direction,  the  energy  of  the  latter  is  the  sum  of 
those  of  the  components. 

6.  Suppose  that  the  two  forks  of  figure  92  had  not  ex- 
actly the  same  period,  say  one  had  100   complete  ,  vibrations 
per  sec.,  the  other  100.25.    What  would  be  the  character  of  the 
vibration  of  the  spot  of  light?     Could  the  conclusions  of  para- 
graph 90  be  modified  to  fit  such  a  case! 

7.  A  pendulum  like  that  of  figure  95  makes  100  complete 
vibrations  per  min.  in  one  direction,  100.25  in  the  perpendicular 
direction.    How  long  would  it  take  to  run  through  the  cycle  of 
changes  portrayed  in  fig.  94? 


CHAPTER  XI. 

94.  Inverse  square  law. — 95.  photometry. — 96.  Rumford  photometer. 
— 97.  Bunsen.  photometer. — 98.  Lummer-Brbdhun  photometer. — 99. 
Light-standards. — 100.  Solid  angle. — 101.  Intrinsic  luminosity. — 102. 
Spectrophotometer. 

94.  Inverse  square  law. — Little  has  been  said  in  this  book 
so  far  about  the  'brightness,  or  intensity,  of  light,  except  for 
the  one  point  that  it  is  proportional  to  the  square  of  the  ampli- 
tude. The  subject  is,  however,  of  great  importance,  not  only 
for  practical  illumination,  but  also  for  the  details  of  experi- 
mental work. 

Suppose  that  S,  figure  96,  is  a  math- 
ematical point,  emitting  light  at  a  steady 
rate  uniformly  in  all  directions.  We  also 
suppose  that  the  medium  exerts  no  ab- 
sorption, which  is  absolutely  true  for  the 
ether,  and  almost  true  for  the  air  and 
other  colorless  gases,  so  far  as  visible 
rays  are  concerned.  Consider  two  spheres, 
with  S  as  center,  of  radii  ra  and  r2.  Evi- 
dently the  same  amount  of  energy  will  pass  in  each  second 
through  each  of  these  spherical  surfaces,  and  therefore  since  tho 
area  of  a  sphere  is  directly  proportional  to  the  square  of  its  ra- 
dius, the  passage  of  energy  per  second  per  square  centimeter  will 
be  inversely  proportional  to  the  square  of  the  radius.  To  put 
the  matter  in  another  form,  suppose  two  white  screens,  each 
1  square  centimeter  in  area,  are  placed  one  50cm.  the  other 
100cm.  from  S,  each  being  turned  so  that  its  face  is  perpen- 
dicular to  the  line  joining  its  center  to  S.  The  nearer  screen 
will  receive  four  times  as  much  light  as  the  more  distant  one. 
and  will  therefore  appear  four  times  as  brightly  illuminated. 
Incidentally,  the  fact  that  the  passage  of  energy  per  second 
varies  inversely  as  the  square  of  the  distance  shows  that  the 
amplitude  of  the  light  varies  simply  inversely  as  the  distance. 

If,  instead  of  a  single  point  source,  there  are  two  points 
or  more,  or  even  an  extended  bright  source  like  a  flame,  com- 
prising a  multiplicity  of  bright  points,  the  inverse  square  law 

(195) 


196  LIGHT 

still  holds  true  provided  that  the  dimensions  of  the  source  are  so 
small  compared  to  its  distance  from  the  screen  that  for  prac- 
tical purposes  we  may  say  that  all  points  of  the  former  are 
equally  distant  from  the  latter.  (This  statement  would  not 
hold  true  if  the  vibrations  in  the  different  emitting  centers  had 
any  special  and  constant  phase-relations,  for  then  interference 
would  take  place  and  the  intensity  of  the  radiation  would  be 
different  in  different  directions.  In  an  actual  source  such  as 
a  flame  or  an  incandescent  filament,  the  vibrations  in  different 
atoms  or  molecules  are  entirely  independent,  the  phase-con- 
stants are  distributed  at  random,  and  in  addition  the  phase  of 
any  particular  vibration  is  no  doubt  subject  to  sudden  abrupt 
changes.  Consequently  interference  in  the  ordinary  sense  can- 
not occur.) 

95.  Photometry. — Instead  of  considering  the  illumination 
of  two  screens  at  different  distances,  produced    by    the    same 
source,  let  us  now  compare  the  illumination  of  the  same  screen 
by  two  different  sources.     By  placing  the  brighter  source  far- 
ther from  the  screen  than  the  fainter  one,  it  is  possible  to  make 
both  sources  produce  the  same  illumination,    and   the    relative 
brightness  of  the  two  sources  can  be  expressed  in  terms  of  the 
two  distances.     The  only  practical  difficulty    lies    in    judging 
when  the  illumination  produced    by    the    two    sources    is    the 
same.     Various  instruments  known  as  photometers,   a  few  of 
which  will  be  described,  have  been  devised  for  this  purpose. 

96.  Rumford    photometer. — The    Rumford    photometer, 
figure  97,  is  a  very  simple  affair.     S  and  S'  are  the  sources 

to  be  compared,  R  a 
cylindrical  rod  shown 
in  cross-section,  and  ss 
a  white  screen.  The 
part  of  the  screen  A  is; 
in  the  shadow  from  S, 
but  is  illuminated  by 
Figure  97  S',  while  the  part  B  is 

in  the  shadow  from  S'  but  is  illuminated  by  S.  The  sources  are 
moved  nearer  to  the  screen  or  farther  away,  keeping  the  edges 
of  the  shadows  in  contact,  till  A  and  B  are  equally  bright.  Let 
d  and  d'  then  be  the  distances  of  S  and  S'  from  the  screen,. 


PHOTOMETERS  197 

while  L  and  L'  represent  the  brightness  of  the  two  sources. 
Since  the  illumination  on  the  screen  is  the  same  from  both 

JL   _!/ 
dJ~~JF 

or 

-=     — 
L'  "~  d" 

This  photometer  is  unsatisfactory  in  practise,  because  it  is  very 
difficult  to  tell  accurately  when  two  illuminated  surfaces  are 
exactly  of  the  same  brightness  unless  they  are  exactly  adjacent 
to  one  another,  without  any  dividing  area  between.  It  is  im- 
possible to  arrange  the  two  shadows  in  this  way  unless  the 
sources  are  very  small.  There  is  always  at  the  edge  of  each 
shadow,  a  region  of  "penumbra,"  or  half-shadow,  where  the 
screen  is  illuminated  by  part  of  one  of  the  sources,  but  hidden 
from  the  rest,  and  the  overlapping  half-shadows  from  the  two 
sources  cause  a  region  of  unequal  illumination  between. 

97.   Bunsen  photometer. — This  is  shown  in  figure  98.   The 
screen  ss  is  placed  between  the  two  sources  S  and  S',  so  that  it 
is  illuminated  by  them 
on    opposite    sides.     A  s, 

greased   spot  is   in   the     s*'~ 
center    of    the    paper 
screen,  and  use  is  made  * 

of  the  fact  that  greased  Figure  98 

paper  transmits  more  light,  and  reflects  less,  than  un- 
greased.  If  no  absorption  occurred  in  either  the  clean  paper 
or  the  greased  portion,  and  if  the  latter  scattered  the  light  as 
effectively  as  the  former,  then  the  grease-spot  would  disappear 
when  the  illumination  is  the  same  on  both  sides.  For,  when 
viewed  from  the  right,  for  instance,  it  would  make  up  in  light 
transmitted  from  S'  what  deficiency  there  was  in  the  light  it 
reflected  from  S,  and  therefore  would  appear  just  as  strongly 
illuminated  as  the  clean  paper  surrounding  it.  Unfortunately, 
there  is  absorption,  and  unequal  absorption,  in  the  two  por- 
tions, and  the  greased  portion  does  not  scatter  as  well  as  the 
ungreased,  but  acts  more  like  a  transparent  medium,  so  that 
the  spot  may  disappear  when  viewed  at  a  certain  angle,  but 


198  LIGHT 

not  when  viewed  at  another.  For  this  reason,  it  is  necessary 
to  view  both  sides  of  the  paper  at  once,  at  the  same  angle.  A 
pair  of  mirrors  are  set  in  an  inclined  position  in  such  a  way 
that  the  observer  stationed  at  E  sees  both  surfaces  of  the 
screen  reflected  in  the  mirrors.  The  screen  and  mirrors,  which 
are  rigidly  connected  together,  are  then  moved  to  right  or  to 
left  between  the  two  sources  until  the  spot  appears  equally 
conspicuous  in  the  two  mirrors.  The  brightnesses  of  the  sources 
are  then  directly  proportional  to  the  squares  of  the  distances 
from  the  screen.  Only  a  small  degree  of  accuracy  is  obtain- 
able with  this  instrument. 

98.  Lummer-Brodhun  photometer. — All  accurate  com- 
parisons of  the  intensities  of  artificial  light-sources  are  made 
with  the  Lummer-Brodhun  photometer,  shown  in  figure  99,  or 
some  modification  of  it.  S  and  S'  are  the  sources  to  be  com- 
pared, and  s  a  two-faced  screen,  made  of  some  white  and 
efficiently  diffusing  material,  such  as  fresh  plaster  of  Paris,  or 

a    fine    quality    of    milk- 
s'    glass.     A  and  B   are  two 

7|\" 


\i/ 

right-angled   glass   prisms, 

so   se^   as   *°   Pr°duce>    Dv 
total  reflection  at  the  hy- 

potenuse,  reflected  images 
Of  the  two  screen-faces. 
P  and  Q  are  another  pair 
of  right-angled  prisms, 
Figure  "  w  i  t  h  t  h  e  i  r  hypotenuse 

faces  cemented  together  by  means  of  Canada  balsam,  after 
part  of  the  face  of  P  has  been  ground  away,  leaving  only  a 
circular  spot  in  the  center  where  actual  contact  occurs.  T  is 
a  short-focus  telescope,  focussed  upon  the  cemented  faces  of  P 
and  Q.  That  part  of  the  light  from  the  right-hand  surface  of 
s  which,  after  reflection  at  B,  strikes  the  cemented  area  where 
P  and  Q  come  together,  passes  through  without  reflection  and 
never  reaches  the  telescope,  but  that  part  which  strikes  the 
edges  is  totally  reflected  into  the  telescope.  Consequently,  if 
the  source  S  were  cut  off  and  S'  alone  functioning,  the  tele- 
scope would  show  an  illuminated  area  with  a  completely  dark 
circular  hole  in  it.  On  the  other  hand,  of  the  light  coming 


LIGHT-STANDARDS  199 

from  the  left-hand  surface  of  s,  illuminated  by  S,  only  that 
part  which  strikes  the  cemented  area  passes  through  to  the 
telescope.  The  field  of  view  then  is  a  circle  illuminated  by  S 
surrounded  by  an  area  illuminated  by  S',  and  since  these  come 
accurately  edge  to  edge  it  is  quite  easy  to  tell  when  the  illumi- 
nation is  the  same.  The  screen,  the  prisms  and  the  telescope 
are  mounted  together  in  a  frame  which  can  be  moved  to  right 
or  left  till  the  inner  circle  vanishes  against  the  equally  illumi- 
nated area  surrounding  it.  Very  accurate  measurements  can 
be  made  with  this  instrument  when  the  lights  to  be  compared 
have  the  same  color.  If  they  differ  much  in  color  an  accurate 
comparison  is  impossible,  for  from  the  nature  of  things  one 
cannot  make  accurate  quantitative  comparisons  of  things  which 
differ  in  quality.  Most  artificial  light-sources,  however,  are 
near  enough  the  same  color  so  that  fairly  trustworthy  com- 
parisons can  be  made. 

99.  Light-standards. — In  measuring  the  brightness  of  a 
source,  some  unit  is  necessary.    The  original  unit  was  the  light 
from  a  spermaceti  candle,  of  a  specified  size  and  burning  at  a 
specified  rate,  and  lights  are  still    rated    in    "candle-power." 
Actual  candles,  however,  even  when  made  and  used  according 
to  specifications,   are  quite  variable,   and  in  practice  a  more 
constant  standard  is  used,  such  as  the  Vernon-Harcourt  lamp, 
taken    as    having    10    candle-power,    or    the    Hefner    lamp,  .9 
candle-power.     Laboratory  tests  are  usually  made  with  an  in- 
candescent lamp  whose  intensity  has  been  standardized  at  the 
Bureau  of  Standards  or  some  similar  laboratory  by  comparison 
with  a  Vernon-Harcourt  or  Hefner  lamp. 

The  "foot-candle"  is  a  unit  of  measurement,  not  for  the 
intensity  of  a  source,  but  for  the  degree  of  illumination  on  a 
surface.  It  is  the  illumination  produced  at  a  distance  of  one 
foot  from  a  source  of  unit  candle-power.  The  degree  of  illumi- 
nation on  a  desk  8  feet  away  from  a  30  candle-power  lamp  is 
30/(8)2  =  30/64  =  .453  foot-candles. 

100.  Solid  angle. — In  order  to  discuss  intelligently  the 
brightness  of  extended  surfaces,  optical  images,  etc.,  it  is  neces- 
sary to  define  a  term  called  "solid  angle."     Suppose  a  cone  of 
rays,  of  any  cross-sectional  shape,  to  emanate  from  a  point  S, 
figure  100.    The  solid-angle  of  this  cone  is  the  area  that  it  cuts 


200 


LIGHT 


Figure   100 


out  on  the  surface  of  a  sphere  of  unit  radius,  or  the  area  that 
it  cuts  out  on  any  sphere  with  S  as  center  divided  by  the  square 

i.^ ^  of    the    radius    of    the    sphere. 

xX  X  Since  the  total  area  of  a  sphere 

\          is  4rrr2  the  total  of  all  solid-an- 
/  \        gles  from  a  point  is  4nr. 

101.  Intrinsic  luminosity. — 
Suppose  we  have  an  extended 
surface  emitting  light,  such  as 
the  surface  of  molten  metal,  or 
of  the  pole  of  a  carbon  arc.  Con- 
sider the  light  emitted,  within 
a  solid-angle  o>,  from  a  small 
area  a,  so  small  that  we  may  regard  it  all  as  lying  at  the  apex 
of  the  solid-angle.  This  amount  of  light  will  evidently  be  pro- 
portional to  the  size  of  the  solid-angle  <o,  and  therefore  may  be 
written  as  equal  to 


The  factor  of  proportionality,  1,  evidently  is  an  indicator  of 
the  brightness  of  the  surface,  without  regard  to  its  size,  and  it 
is  given  the  name  " intrinsic  luminosity."  It  may  be  defined 
in  words  as  the  amount  of  light  emitted  per  unit  area  per  unit 
solid-angle. 

We  shall  now  prove  that,  when  a  real  image  is  formed  by 
a  lens,  the  intrinsic  luminosity  of  the  image  is  the  same  as  that 
of  the  object,  except  for  losses  of  light  due  to  reflection,  absorp- 
tion, etc.  In  figure  101  let  a  be  any  very  small  area  on  the 
object  0  and  b  the  corresponding  area  on  the  image  I.  The 


Figure    101 

amount  of  light  from  a  going  to  form  b,  apart  from  the  above- 
mentioned  losses,  is  the  amount  lying  within  the  solid-angle  of 
the  cone  which  enters  the  lens.  The  value  of  this  solid-angle  is 


INTRINSIC  LUMINOSITY  201 

where  A  stands  for  the  area  of  the  lens  and  u  for  the  distance 
of  the  object  from  it.  If  I  is  the  intrinsic  luminosity  at  a,  the 
amount  of  light  from  a  entering  the  lens  is 

lAa 


Let  I'  be  the  intrinsic  luminosity  of  the  image  at  b,  v  the  dis- 
tance of  the  image  from  the  lens,  and  a/  the  solid-angle  of  the 
cone  which  goes  to  form  b  and  the  equal  cone  which  emerges 
from  b  on  the  right,  then 


and  the  amount  of  light  going  to  form  b  is 

_PAb 

If  no  light  were  lost,  we  could  then  write 

IAa__  I'Ab 
u2   "       v34 

The  dimensions  of  any  part  of  an  image  are  to  those  of  the 
corresponding  part  of  the  object  as  v  is  to  u,  and  therefore  the 
areas  will  be  in  the  ratio  v2  to  u2,  so  that 

«-  i 

..  —  ., 

-2*.  u'    v* 

Consequently 

1  =  1' 

This  conclusion  is  at  first  sight  surprising,  for  it  seems  to  indi- 
cate— first  that  the  brightness  of  the  image  per  unit  area  is 
independent  of  the  distance  of  the  object, — second  that  it  is 
independent  of  the  diameter  of  the  lens.  Distance  of  the  object 
does  indeed  have  no  effect,  for  although  the  lens  receives  less 
light  when  the  object  is  farther  away,  that  light  is  distributed 
over  an  image  whose  area  is  smaller  in  the  same  proportion. 

As  to  the  effect  of  the  diameter  of  the  lens,  one  must 
remember  that  intrinsic  luminosity  is  defined  for  unit  solid- 
angle,  as  well  as  for  unit  area.  A  large  lens  does  in  fact  send 
more  light  to  form  the  image,  but  since  the  solid-angle  is 


202  LIGHT 

increased  in  the  same  proportion  the  intrinsic  luminosity 
remains  the  same.  If  the  image  is  cast  on  a  photographic  plate, 
where  the  photographic  action  depends  only  upon  the  total 
amount  of  light  received  per  unit  area  of  plate,  entirely  with- 
out regard  to  the  size  of  the  light-cone,  it  is  of  course  advan- 
tageous to  use  a  lens  of  large  diameter.  The  same  is  true  if 
the  image  is  cast  on  a  diffusing  screen,  like  white  paper  or  plas- 
ter of  Paris,  for  such  materials  do  not  allow  the  light  to  remain 
in  a  cone  of  the  same  diameter  as  that  in  which  it  comes  to  the 
screen,  but  re-emit  it  in  all  directions.  In  either  of  the  above 
cases,  the  lens  should  be  as  large  in  diameter  as  is  consistent 
with  moderate  freedom  from  spherical  aberration,  astigmatism, 
etc.,  if  a  bright  image  is  desired. 

L 


Figure  102 

But  when  an  image  is  viewed  directly  by  the  eye,  as  in 
figure  102,  the  amount  of  light  entering  the  eye  is  limited  by 
the  size  of  the  pupil,  and  all  the  light  which  fails  to  enter  the 
pupil  is  wasted,  so  far  as  vision  is  concerned.  Consequently, 
the  brightness  of  the  final  image  upon  the  retina  is  not  affected 
by  an  increase  or  decrease  in  the  diameter  of  the  lens,  so  long 
as  the  cone  of  light  remains  large  enough  to  completely  fill  the 
pupil.  If  the  pupil  were  not  completely  filled,  the  intrinsic 
brightness  of  the  image  on  the  retina  would  still  be  the  same, 
but  the  mental  impression  of  brightness  would  be  reduced.  For 
this  impression,  like  photographic  action,  depends  upon  the 
amount  of  light  received  without  regard  to  the  size  of  the  cone 
of  light. 

In  spite  of  all  that  has  been  said,  there  is  a  great  advan- 
tage in  the  use  of  large-diameter  lenses  as  the  objectives  for 
telescope.  In  the  first  place,  a  large-diameter  objective  permits 
the  use  of  a  very  short-focus  eyepiece,  and  therefore  a  high 
magnifying-power,  without  reducing  the  emerging  cone  of  light 
so  much  that  it  does  not  fill  the  pupil.  In  the  second  place, 
we  have  seen  that  the  image  of  a  mathematical  point  is  not  a 
point,  but  a  small  disc  surrounded  by  series  of  fainter  concentric 


INTRINSIC  LUMINOSITY  203 

rings  (see  sections  72  and  73),  and  that  the  diameter  of  disc  and 
rings  become  smaller  when  the  diameter  of  the  objective  becomes 
larger.  A  fixed  star  (what  follows  does  not  apply  to  planets) 
may  be  many  times  larger  than  the  sun,  but  its  distance  is  so 
great  that  in  every  case  what  we  call  its  "geometrical"  image 
— the  image  as  it  would  be  if  there  were  no  such  thing  as 
diffraction — is  hardly  more  than  a  point,  being  much  smaller 
than  the  central  disc  of  the  diffraction  pattern.  "We  are  there- 
fore obliged  to  regard  a  star  as  equivalent  to  a  point-source, 
whose  image  is  the  diffraction-pattern  itself.  Since  the  diameter 
of  the  central  diffraction  disc  is  inversely  proportional  to  the 
diameter  of  the  objective,  it  is  clear  that  the  star-image  will 
have  a  much  greater  intrinsic  luminosity  with  a  large  than 
with  a  small  lens.  On  the  other  hand,  when  an  object  is  large 
enough  or  close  enough  so  that  its  geometrical  image  has  per- 
ceptible size,  although  each  point  of  the  image  is  made  .up  of 
diffraction-disc  and  rings,  the  only  effect  of  these  is  to  extend 
the  edges  of  the  image  very  slightly  indeed,  and  the  effect  of 
diffraction  upon  the  area  of  the  total  image  is  relatively  negli- 
gible. Suppose  that  a  telescope  be  pointed  toward  a  star  in 
daylight.  Since  the  diameter  of  the  objective  exceeds  greatly 
that  of  the  pupil  of  the  eye,  the  intrinsic  luminosity  of  the 
star  is  much  greater  than  when  seen  with  the  naked  eye;  but 
that  of  the  sky,  an  extended  area,  is  not  increased  in  the  least 
degree.  Consequently,  it  is  possible  to  see  with  a  large  tele- 
scope stars  totally  invisible  to  the  naked  eye,  even  in  full  day- 
light. 

The  principle  of  equality  of  image  and  object  so  far  as 
intrinsic  luminosity  is  concerned,  proved  above  (with  reserva- 
tions as  to  loss  of  light  by  reflection  and  absorption)  for  a  real 
image  formed  by  a  lens,  can  be  just  as  easily  proved  for  a  vir- 
tual image,  or  for  an  image  formed  by  a  mirror.  The  only 
advantage  in  using  opera-glasses,  reading  glasses,  microscopes, 
etc.,  so  far  as  extended  objects  are  concerned,  is  the  magnifica- 
tion in  size.  The  brightness  is  never  increased,  but  rather 
somewhat  diminished  by  the  unavoidable  losses. 

It  is  easy  to  prove  also  that,  except  for  atmospheric  ab- 
sorption, an  object  will  appear  with  the  same  intrinsic  bright- 
ness at  any  distance,  provided  the  pupil  does  not  dilate  or 
contract.  For  in  figure  101  the  lens  L  can  just  as  well  repre- 


204  LIGHT 

sent  the  lens  of  the  eye  as  any  other  lens,  with  the  image  I 
formed  on  the  retina.  The  proof  then  follows  word  for  word 
as  given  above. 

102.  Spectrophotometer. — It  has  already  been  mentioned 
that  accurate  comparisons  of  the  brilliancy,  or  candle-power, 
of  light-sources  can  be  made  only  when  they  have  about  the 
same  color.  To  compare  the  intensities  of  a  reddish  and  a 
bluish  light  would  be  much  like  comparing  the  intensities  of  a 
deep  bass  musical  note  and  a  shrill  treble  one.  Precise 
measurements  can  never  be  made  except  when  the  objects  com- 
pared, though  differing  in  quantity,  are  the  same  in  quality. 
It  is  true  that  a  special  form  of  instrument,  known  as  the 
"flicker-photometer,"  has  been  used  with  some  success  in  com- 
paring lights  somewhat  divergent  in  color,  but  the  indications 
given  by  it  are  now  considered  unreliable,  and  their  interpre- 
tation is  in  any  case  somewhat  doubtful. 

The  only  completely  satisfactory  method  for  such  a  problem 
is  to  form  the  spectra  of  the  two  sources  side  by  side  and  then 
make  a  comparison,  color  by  color,  for  a  number  of  different 

parts    of  the  spectrum.     Several    de- 

'B  vices  for  accomplishing  this,  known  as 

' '  spectrophotometers "  are  now  in  use. 
The  results  of  the  comparison  must  be 
given  in  the  form  of  a  table,  or  bet- 

ter  still  a  curve,  with  wavelength  as 

abscissa  and  the  corresponding  ratio 
of  brilliancy  in  the  two  spectra  as  or- 

dinate.  Thus  a  curve,  such  as  that  in  figure  103  would  indicate 
that  the  source  A  is  relatively  stronger  than  source  B  in  the 
middle  of  the  spectrum,  about  the  green. 

Problems. 

1.  What  is  the  value  of  the  solid  angle  included  between 
two  walls  and  the  floor  of  a  room? 

2.  Find  the  illumination  on  a  desk  10  ft.  from  a  60  candle- 
power  lamp.     Suppose  the  desk  were  inclined  in  such  a  way 
that  a  perpendicular  to  it  made  an  angle  A  with  a  line  drawn 
from  it  to  the  lamp.     What  would  then  be  the  illumination! 

3.  Given  that  the  illumination  of  the  earth  in  full  moon- 
light is  .02  foot-candles,  and  that  the  distance  of  the  moon  is 
235000  miles,  find  its  candle-power. 


CHAPTER  XII. 

103.  Transverse  and  longitudinal  waves. — 104.  Double  refractiors. 
— 105.  Polarization  of  the  O  and  B  light. — 106.  Wave-surface  in  doubly- 
refracting  crystals. — 107.  The  lateral  displacement  of  the  E  ray. — 108. 
Special  cases  of  double  refraction. — 109.  Tourmaline. — 110.  Biaxial 
crystals. — 111.  Polarization  by  reflection. 

103.  Transverse  and  longitudinal  waves. — We  have  al- 
ready spoken  of  the  distinction  between  longitudinal  and  trans- 
verse waves,  but  have  left'  unsettled  the  question  to  which  of 
these  types  light-waves  belong.  A  decision  has  so  far  been  un- 
necessary, because  everything  we  have  said  about  light  up  to 
this  point  would  apply  equally  well  whether  the  waves  were 
longitudinal  or  transverse.  For  instance,  reflection,  refraction, 
interference,  diffraction,  etc.,  could  occur  equally  well  with 
either. 

It  is  manifestly  impossible  to  see  a  light-wave,  in  the  sense 
that  we  see  water-waves,  or  waves  in  a  string.  "We  see  by 
means  of  light,  but  we  do  not  see  light  itself.  Therefore  the 
character  of  the  waves  must  be  determined  by  indirect,  rather 
than  direct,  means.  The  most  important  distinction  between 
the  two  types  is  this,  that  a  train  of  longitudinal  waves  is  com- 
pletely specified  when  we  have  stated  its  wavelength,  ampli- 
tude, phase,  and  direction  of  propagation,  while  a  train  of 
transverse  waves  is  not.  To  explain  more  fully,  imagine  a  beam 
of  monochromatic  light  travelling  from  north  to  south,  whose 
amplitude,  wavelength  and  phase  are  known.  If  the  waves 
are  longitudinal,  nothing  more  can  be  said  about  this  beam, 
and  any  other  beam,  travelling  in  the  same  direction,  with  the 
same  amplitude,  wavelength  and  phase,  would  be  indistinguish- 
able from  it.  On  the  other  hand,  if  the  waves  are  transverse, 
there  is  still  a  possibility  that  the  two  beams  should  be  quite 
different.  For  instance,  the  vibrations  in  the  first  might  be  up 
and  down,  while  those  in  the  second  were  east  and  west. 

Evidently  then,  if  light  waves  are  transverse,  there  should 
exist  some  distinguishing  characteristic  of  a  beam  of  light, 
other  than  wavelength,  amplitude,  phase,  and  direction  of  pro- 
pagation, and  having  to  do  with  a  direction  at  right  angles  to 
the  direction  of  propagation,  if  we  could  only  find  the  suitable 

(205) 


206  LIGHT 

means  of  detection.  But  we  should  hardly  expect  ordinary 
natural  light,  such  as  comes  directly  from  the  sun  or  a  flame, 
to  show  this  characteristic  even  if  it  exists ;  for  there  would  be 
no  reason  why  in  such  light  one  direction  of  vibration  would 
predominate  over  another.  It  would  be  much  more  probable 
that  all  possible  directions  of  vibration  would  be  represented 
to  about  the  same  extent,  so  that  the  beam  would  not  show  any 
peculiar  properties  in  any  direction.  Consequently,  in  seeking 
such  a  characteristic  as  we  have  mentioned  as  possible,  which 
would  prove  light-waves  to  be  transverse,  we  ought  to  experi- 
ment, not  with  light  coming  directly  from  any  original  source, 
but  rather  with  light  which  has  suffered  reflection,  refraction, 
or  some  other  action  which  might  cause  one  plane  through  the 
direction  of  propagation — for  instance  the  plane  of  incidence  in 
the  case  of  reflection — to  have  a  particular  importance  for  the 
beam. 

104.  Double  refraction. — In  1690  Huyghens  discovered  a 
characteristic  of  light,  such  as  we  have  been  speaking  about, 
which  we  regard  as  a  definite  proof  that  light-waves  are  trans- 
verse. Huyghens  himself  did  not  draw  this  conclusion,  for  at 
that  time  it  had  always  been  assumed  without  question  that 
the  waves  were  longitudinal,  like  those  of  sound,  and  the  pos- 
sibility of  their  being  transverse  had  never  been  suggested. 
Huyghens'  discovery  was  concerned  with  the  phenomenon 
known  as  "double  refraction,"  which  is  shown  by  many  crys- 
tals, to  the  most  remarkable  extent  by  the  crystal  known  as 
calcite,  or  Iceland  spar.  Before  Huyghens'  experiment  can  be 
understood,  it  will  be  necessary  to  explain  some  of  the  facts 
about  crystal  structure  and  the  nature  of  double  refraction. 

Every  crystal  possesses  certain  regularities  of  structure, 
on  account  of  which  it  can  be  easily  split  in  certain  directions. 
Calcite,  for  instance,  can  be  readily  split  into  any  one  of  the 
forms  shown  in  figure  104,  which  differ  in  linear  dimensions, 
but  have  exactly  the  same  angles.  The  important  thing  about 
crystal  structure  is  that  the  angles  are  fixed  and  invariable, 
while  the  dimensions  of  the  faces  may  be  anything.  In  any  one 
of  the  three  rhombohedral  forms  shown  in  the  figure  for  cal- 
cite, each  face  has  two  angles  of  101°  55'  and  two  of  78°  5'. 
In  each  rhombohedron,  there  are  two  opposite  corners  where 
three  obtuse  angles  come  together,  as  at  A.  A  line  drawn 


DOUBLE  REFRACTION 


207 


through  one  of  these  corners,  equally  inclined  to  the  three  faces 
that  meet  there,  or  any  line  parallel  to  it,  is  called  the  optic 
axis.  Note  that  the  optic  axis  is  defined  only  by  its  direction, 
and  any  line  having  that  direction  may  be  called  the  optic  axis. 


Figure   104 

When  a  pencil  of  light  enters  such  a  rhomb  of  calcite  it  is 
separated  into  two  parts,  so  that  there  are  two  refracted  beams, 
instead  of  only  one  as  in  the  case  of  glass  and  other  non-crys- 
talline media.  One  of  the  beams  obeys  the  ordinary  laws  of 
refraction,  the  ratio  of  the  sines  of  the  angles  of  incidence  and 
refraction  being  always  the  same,  no  matter  what  the  angle  of 
incidence  may  be.  It  is  therefore  called  the  "ordinary"  ray, 
and  we  shall  represent  it  by  the  letter  O,  for  brevity.  The  other 
does  not  follow  the  ordinary  laws  of  refraction.  The  ratio 
of  the  sines  changes  as  the  angle  of  incidence  changes,  showing 
that  this  beam  travels  through  the  calcite  with  different 
velocities  in  different  directions,  this  being  due,  no  doubt,  to 
the  regular  structure  of  the  crystal.  For  this  reason  it  is  called 
the  "extraordinary"  ray  and  we  shall  rep- 
resent it  by  E.  If  we  wish  to  speak  of 
-* — f  an  index  of  refraction  for  the  extraordi- 
nary ray,  it  must  be  understood  that  that 
index  is  not  a  constant  for  any  given  wave- 
Figure  los  length,  but  is  variable  with  the  angle 
of  incidence.  Even  when  the  incidence  is  normal,  when  there 
should  be  no  refraction  according  to  the  ordinary  law,  the 
E-ray  is  in  general  refracted,  as  illustrated  in  figure  105. 
ABCD  is  the  sectional  outline  of  a  crystal  of  calcite.  The 
incident  ray  PQ,  even  when  it  strikes  the  surface  normally  as 


208  LIGHT 

i 

in  the  figure,  is  divided  into  two  rays,  one  of  which  PQRT  (the 
O-ray)  goes  straight  through  without  bending,  while  the  other, 
PQSU  (the  E-ray),  is  deflected  upon  entering  the  calcite  and 
emerges  from  it  parallel  to  the  incident  ray  but  with  a  lateral 
displacement.  Consequently,  if  an  object  close  to  the  rhomb 
be  viewed  through  the  latter,  two  images  will  be  seen.  If  the 
object  is  very  far  away,  however,  only  a  single  image  will  be 
seen ;  for  the  two  emergent  rays  are  parallel,  with  only  a  small 
relative  displacement,  and  for  an  object  very  far  away  a  small 
displacement  does  not  change  its  apparent  position  by  an  ap- 
preciable amount.  Therefore  the  two  images  coincide  and  ap- 
pear as  a  single  image.  Another  explanation  is  as  follows:  If 
the  object  is  far  away,  the  incident  waves  are  practically  plane, 
and  therefore  the  two  emergent  beams  have  waves  that  are 
practically  plane,  and  parallel  to  one  another,  and  we  have 
already  found  that  all  plane  and  parallel  waves  are  focussed  at 
the  same  point  on  the  retina.  Conversely,  we  can  regard  the 
fact  that  only  a  single  image  of  a  distant  object  can  be  seen 
through  a  block  of  calcite  as  proof  that  the  two  emergent  rays 
coming  from  a  single  incident  ray  are  parallel  to  one  another. 

The  two  images  of  a  near  object  seen  through  calcite, 
formed  respectively  by  the  O-ray  and  the  E-ray,  are  called  the 
ordinary  and  the  extraordinary  images.  If  the  crystal  is 
turned  about  the  incident  ray  as  an  axis,  the  0-image  remains 
still  and  the  E-image  revolves  about  it.  Let  us  define  a  plane 
which  is  parallel  to  the  optic  axis  and  perpendicular  to  the 
face  through  which  the  light  enters  as  the  principal  plane  of 
that  face.  A  line  joining  the  centers  of  the  two  images  lies  in 
the  principal  plane.  The  images  are  equally  bright,  and  the 
light  from  them  seems  exactly  the  same  to  the  eye.  A  super- 
ficial examination  reveals  no  difference  between  them  except 
for  the  displacement  of  the  E-image  in  the  principal  plane, 
and  the  further  fact  that  it  seems  slightly  farther  away  than 
the  0-image.  But  note  that  the  two  refracted  beams  have 
undergone  a  process  (transmission  through  a  crystal)  which 
has  caused  a  particular  plane  (the  principal  plane)  to  have 
a  peculiar  importance  for  both.  Consequently,  in  accordance 
with  our  former  discussion,  if  the  waves  are  transverse  we  are 
more  likely  to  get  evidence,  for  it  in  these  two  beams  than  in 


POLARIZATION 


209 


the  original  incident  beam,  where  one  direction  of  vibration 
has  no  reason  to  be  preferred  over  another. 

105.  Polarization  of  the  0  and  E  light. — Such  evidence  is 
furnished  in  the  experiment  of  Huyghens,  mentioned  above, 
which  consisted  in  viewing  an  object  through  two  crystals  of 
calcite.  If  the  crystals  are  equally  thick  and  are  similarly 
placed  so  that  their  optic  axes  are  parallel,  as  in  figure  106, 
the  phenomenon  observed  is  just  the  same 
as  with  only  one,  except  that  the  displace- 
ment between  the  images  is  doubled.  For 
the  two  are  equivalent  to  a  single  crys- 
106  tal  of  double  thickness.  But  if  one  crys- 

tal is  turned  about  the  incident  ray  as  axis,  while  the  other 
is  held  still,  there  are  in  general  four  images,  though  for 
certain  positions  of  the  moving  crystal  two  of  them  disappear, 
and  in  one  position  only  one  is  visible.  The  production  of 
four  images  is  not  surprising,  for  one  should  expect  that  the 


6  H  I  J 

Figure   107 

second  crystal  would  divide  each  of  the  two  beams  that  enter 
it  into  two  more,  but  the  changes  in  intensity  that  the  images 
undergo  when  the  second  crystal  is  rotated,  and  the  dis- 
appearance of  some  of  them  in  certain  positions,  require 
explanation.  These  changes  should  be  understood  from  the 
series  of  eight  diagrams  in  figure  107.  In  all  of  them  thr,- 
plane  of  the  paper  is  supposed  perpendicular  to  the  incident 
ray,  and  the  little  circles  show  the  positions  of  the  images.  In 
diagram  A,  aX  represents  the  intersection  of  the  principal 


210  LIGHT 

plane  for  the  first,  or  stationary,  crystal  with  the  plane  of  the 
paper.  aY  that  for  the  second,  or  moving,  crystal,  the  two  mak- 
ing in  this  diagram  an  angle  a  which  is  less  than  45°.  Four 
images  are  seen,  of  which  a  and  d  are  brighter  than  b  and  c. 
If  the  angle  a  be  increased,  a  and  d  become  fainter  and  b  and 
c  brighter,  till  when  a  =  45°,  as  in  diagram  B,  all  four  are 
equally  bright.  If  a  is  still  further  increased,  a  and  d  con- 
tinue to  grow  dimmer,  b  and  c  brighter,  and  when  a  =  90° 
(C)  a  and  d  disappear  and  b  and  c  attain  maximum  bright- 
ness. With  a  further  increase  in  a,  a  and  d  reappear  and 
grow  brighter,  while  b  and  e  diminish,  till  when  a  =  135° 
they  are  all  equally  bright,  and  when  a  =  180°  a  and  d  have 
reached  maximum  brilliancy,  b  and  c  vanished  completely. 
Meanwhile,  a  and  b  have  maintained  their  positions,  but  c  has 
swung  about  a  as  a  center,  and  d  about  b,  so  that  in  the  180° 
position  (G)  the  only  remaining  images,  a  and  d,  fall  in  the 
same  place  and  appear  as  a  single  image.  Similar  changes  go 
on  if  a  is  increased  beyond  180°.  H  shows  the  appearance 
when  a  =.  225°  (the  four  images  equally  bright),  I  that  when 
a  =  270  (a  and  d  vanish,  b  and  c  very  bright),  and  J  when 
a  =  275°  (all  equally  bright).  If  a  —  360  it  is  the  same 
as  if  a  =  0,  a  and  d  would  have  maximum  brightness  and  be 
at  the  maximum  distance  apart,  while  b  and  c  would  be  gone. 

These  phenomena  can  all  be  well  explained  if  we  assume 
— first,  the  transverse  nature  of  light  waves: — second,  a  cer- 
tain property  of  crystals  in  regard  to  the  velocity  of  light.  A 
non-crystalline  material  like  glass  has  properties  which  are  the 
same  in  any  direction,  and  light  therefore  travels  through  it 
with  the  same  velocity,  no  matter  what  may  be  the  direction 
of  propagation  or  the  plane  in  which  the  vibrations  occur. 
Crystals,  on  the  other  hand,  have  decidedly  different  proper- 
ties in  different  directions.  Not  only  do  they  split  easily  into 
layers  in  certain  definite  planes,  but  also  such  properties  as 
heat-conductivity  and  the  elastic  and  electrical  constants  are 
different  according  to  the  direction  in  which  they  are  measured. 
Consequently,  taking  the  optic  axis  as  an  axis  of  symmetry,  it 
is  likely  enough  that  waves  whose  vibrations  lie  in  the  principal 
plane  would  be  transmitted  with  a  different  velocity  from  those 


POLARIZATION  21 1 

with  vibrations  perpendicular  to  that  plane,  and  this  is  the 
second  assumption  mentioned  at  the  head  of  this  paragraph. 
Then,  if  a  beam  of  natural  light,  which  presumably  has  vibra- 
tions equally  in  all  directions,  falls  upon  a  crystal,  the  latter 
will  automatically  resolve  it  into  two  component  beams,  the 
vibrations  of  which  are  in  one  parallel,  in  the  other  perpen- 
dicular, to  the  principal  plane.  These  will  become  separated 
from  one  another  precisely  because  one  travels  faster  than  the 
other,  so  that  when  they  emerge  from  the  crystal  we  shall  have 
instead  of  a  single  beam  in  which  all  directions  of  vibrations 
are  equally  represented,  two  beams  in  each  of  which  the  vibra- 
tions are  confined  to  a  plane,  these  planes  for  the  two  beams 
being,  however,  mutually  perpendicular.  Light  restricted  to  a 
single  plane  of  vibration  is  said  to  be  polarized,  or  more 
specifically,  plane  polarized,  so  that  both  the  0  and  E  beams 
are  polarized,  but  not  in  the  same  plane.  Both  beams  are 
equally  bright  because  the  incident  light  (unpolarized)  had 
presumably  no  excess  of  vibrations  in  any  particular  plane. 
Measurements  of  refraction  show  that  the  E-light  travels  faster 
in  the  calcite  than  the  0-light,  but  there  is  no  obvious  reason 
to  decide  whether  vibrations  in  the  principal  plane  or  perpen- 
dicular thereto  should  be  propagated  the  faster.  Consequently 
we  are  at  a  loss  to  know  whether  the  0-light  has  its  vibrations 
in  the  principal  plane  and  the  E-light  perpendicular  thereto, 
or  vice  versa.  "We  avoid  the  dilemma  by  simply  „ 

saying  that  the  0-light  is  polarized  in  the  prin- 
cipal plane,  the  E-light  polarized  perpendicular 
to  the  principal  plane.  These  statements  should 
be  regarded  as  a  definition  of  the  plane  of 
polarization,  leaving  as  a  matter  for  future  dis- 
cussion the  question  whether  the  plan  of  polari- 
zation coincides  with  the  plane  of  the  vibra- 
tions, or  is  perpendicular  to  it. 

Now,  suppose  that  our  two  equally  bright 
polarized  beams,  obtained  by  passage  through  * 

a  single  crystal,  strike  a  second  crystal  whose 
principal    plane    makes    an    angle    a    with    that    of    the    firsts 
In   figure    108,    oX   represents   the   position   of   the    principal 
plane    for   the    first    crystal,    oY    that    for   the    second.     Two 
equal    vectors    are    laid    off,    op    representing    the    amplitude 


212  LIGHT 

of  the  0-ray  as  it  comes  from  the  first  crystal,  eq  that  of  the 
E-ray.     The  vectors  are  drawn  in  the  directions  of  the  respec- 
tive planes  of  polarization,  because  we  are  not  certain  as  to  the 
actual    directions    of    vibration.     Now    light    whose    plane    of 
polarization  is  either  parallel  or  perpendicular  to  oX  cannot  as 
such  pass  through  the  second  crystal,  but  only  light  whose  plane 
of  polarization  is  parallel  or  perpendicular  to    oY.     We    have 
seen,  however,  in  section  92,  that  a  simple  harmonic  motion  of 
amplitude  op  is  equivalent  to  two  other    simple    harmonic  mo- 
tions of  the  same  period  and  phase,  having  amplitudes  oa  and 
oc,  gotten  by  the  rules  of    vector    resolution.     Therefore,    the 
0-beam  from  the  first  crystal  will  be  automatically  broken  up 
by  the  second,  and  will  pass  through  the  latter  as  a  beam  of 
amplitude  oa,  polarized  in  the  principal  plane,  and  a  beam  of 
amplitude  oc,  polarized  perpendicular  to  the    principal    plane. 
The  former  of  these  forms  the  image  a  of  figure  107,  the  latter 
the  image  c.     In  quite  the  same  way  the  E-beam  from  the  first 
crystal,  represented  by  the    vector    eq,    will,    on    entering    the 
second  crystal,  give  rise  to  a  beam  of  amplitude  eb,  polarized 
in  the  principal  plane  and  responsible  for  the  image  b  of  figure 
107,  and  one  of  amplitude  ed,  polarized  perpendicular   to    the 
principal  plane  and  responsible  for  the  image  d  of  figure  107. 

The  relative  intensities  of  the  four  images  a,  b,  c,  and  d  of 
figure  107  can  be  learned  from  the  amplitudes  of  the  vectors 
oa,  oc,  eb  and  ed,  figure  108.  If  A  represents  the  amplitude  op 
or  oq  (representing  the  equal  0  and  E  beams  coming  from  the 
first  crystal)  then 

oa  =  ed  =  A.  cos  a      and      oc  =  eb  =  A.  sin  a 
Since  the  intensity  of  a  beam  of  light  is  proportional    to    the 
square  of  the  amplitude,  the  intensities  of  the  images  a,  b,  c, 
and  d  of  figure  107  are  then  in  the  continued  proportion 

a  :  b  :  c  :  d  =  cos2  a  :  sin2  a  :  sin2  a  :  cos2  a 
Therefore,  when  a  =  90°  or  270°,  a  and  d  vanish  while  b 
and  c  have  maximum  brightness,  and  conversely  when  a  = 
0  or  180°,  and  all  are  equally  bright  when  a  =  45°,  135°, 
225°,  or  315°.  These  being  exactly  the  relations  of  brightness 
which  experiment  shows  to  exist,  we  may  regard  it  as  definitely 
proved  that  light-waves  are  transverse,  and  that  certain  crystals 
have  the  power  to  divide  up  an  unpolarized  beam  of  light  into 
two  component  beams  polarized  at  right-angles  to  one  another. 


WAVE-SURFACE  IN  CRYSTALS  213 

106.  Wave-surface    in    doubly-refracting    crystals. — The 
fact  that  the  ordinary  ray  obeys  the  regular  laws  of  refraction 
shows  that  light  polarized  in  the  principal  plane  travels  with 
the  same  velocity  for  all  directions  of  propagation,  that  is,  for 
all  directions  of  the  ray.    On  the  other  hand,  the  failure  of  the 
extraordinary  ray  to  follow  these    same    laws    shows    just    as 
surely  that  light  polarized  perpendicular  to  the  principal  plane 
does  not  have  the  same  velocity  for  different  directions  of  the 
ray.     Careful   measurements   of   the   angles  of   refraction   for 
various  angles  of  incidence  show  that  velocity  varies  with  ray- 
direction  in  a  manner  that  can  best  be  explained  as  follows: 

Suppose  that,  at  some  point  in  the  interior  of  a  block  of 
glass,  a  disturbance  of  the  molecules  takes  place,  of  the  kind 
which  produces  light.  The  light  would  pass  out  from  this 
point  in  wavefronts  each  of  which  would  be  a  perfect  sphere, 
for  the  velocity  would  be  the  same  for  all  directions.  If  the 
same  sort  of  disturbance  were  to  occur  in  the  interior  of  a 
block  of  calcite,  each  wavefront  would  no  longer  be  a  sphere 
simply,  but  a  double  surface  consisting  of  a  sphere  and  an  ellip- 
soid of  revolution.  The  optic  axis  through  the  center  is  the 
axis  of  the  ellipsoid,  and  ellipsoid  and  sphere  are  tangent  to 
one  another  where  this  axis  pierces  both  surfaces.  The  spheri- 
cal part  is  the  wavefront  for  vibrations  whose  plane  of  polari- 
zation is  a  radial  plane,  a  plane  containing  ray  and  optic  axis. 
The  ellipsoid  is  the  wavefront  for  vibrations  polarized  perpen- 
dicular to  the  radial  planes.  Therefore  the  sphere  accounts 
for  the  ordinary  and  the  ellipsoid  for  the  extraordinary  light. 
In  calcite,  the  ellipsoid  is  oblate,  and  lies  outside  the  sphere, 
and  therefore  the  E-wave  travels  faster  than  the  0-wave,  but 
in  some  crystals  it  is  prolate  and  lies  inside,  and  the  0-wave 
travels  the  faster.  In  the  direction  of  the  optic  axis,  both 
waves  travel  with  the  same  velocity,  and  double  refraction  fails. 
Only  a  single  image  can  be  seen  from  light  passing  through  the 
crystal  parallel  to  the  optic  axis. 

107.  The  lateral  displacement  of  the  E-ray. — We  can  now 
explain  why  the  extraordinary  ray  receives  a  lateral  displace- 
ment in  going  through  a  block  of  calcite   (not  parallel  to  the 
optic  axis)  even  when  the  angle  of  incidence  is  zero.     Let  AB, 
figure  109,  represent  a  plane    surface    separating    a    block    of 
calcite  (left)  from  air  (right),  and  let  XY  be  a  series  of  wave- 


214 


LIGHT 


fronts  advancing  in  the  direction  of  the  arrow,  perpendicular 
to  the  surface.    The  optic  axis  is  parallel  to  Oc,  in  the  plane  of 

the  paper.  According  to  Huyghen's 
principle,  each  point  in  the  surface 
will,  when  a  wavefront  strikes  it, 
become  the  center  of  wavelets  that 
will  advance  into  the  calcite,  and 
the  tangent  to  these  wavelets  con- 
stitutes the  wavefront  in  the  crys- 
tal. Wavelets  are  drawn  in  the 
figure  from  three  centers,  a,  b,  and 
c,  .and  each  shows  in  the  diagram  as 
a  semi-circle  and  a  semi-ellipse, 
these  being  the  sections  of  the  wave- 
surface  with  the  plane  of  the  paper. 
Figure  109  If  the  incident  wavefront  consisted 

only  of  light  polarized  in  the  plane  of  the  paper,  the  semi-ellipse 
would  be  omitted,  for  the  plane  of  the  paper  is  a  radial  plane 
for  the  wavesurface,  and  there  would  be  no  extraordinary  wave. 
If  the  incident  light  were  polarized  perpendicular  to  the  plane 
of  the  paper,  the  semi-circle  would  be  omitted,  and  there  would 
be  no  ordinary  wave.  But  if  the  incident  light  is  either  un- 
polarized,  or  polarized  in  any  other  plane,  both  circle  and 
ellipse  are  represented,  and  both  rays  exist.  The  refracted 
wavefront  will  then  be  double,  consisting  of  a  plane  ukv,  tan- 
gent to  the  circles  and  a  plane  sit  tangent  to  the  ellipses.  These 
two  are  parallel  to  one  another,  but  the  E-wavefront  (sit)  has 
advanced  farther  than  the  0-wavefront  (ukv)  and  has  received 
a  lateral  shift,  which  increases  as  the  wavelets  progress,  until 
the  second  face  of  the  calcite  is  reached.  Then  we  can  again 
apply  Huyghens'  principle  to  find  the  wavefronts  refracted  out 
into  the  air,  remembering  that  here  all  waves  travel  with  the 
same  velocity,  whatever  may  be  the  plane  of  polarization.  Of 
course,  the  terms  ''ordinary"  and  ' ' extraordinary "  apply  only 
to  light  within  a  crystal  or  other  doubly-refracting  material. 
Evidently,  the  ordinary  wavefronts  will  emerge  into  the  air  as 
shown  at  UV  and  the  extraordinary  as  at  ST.  The  latter  will 
have  gained  distance  and  suffered  a  permanent  lateral  displace- 
ment, but  neither  the  distance  gained  nor  the  lateral  displace- 
ment increases  any  more. 


SPECIAL  CASES 


215 


In  the  ordinary  light,  what  we  call  the  "ray,"  or  the  line 
along  which  the  light  advances,  is  perpendicular  to  the  wave- 
front,  because,  the  secondary  wavelets  being  spherical,  the  line 
drawn  from  a  center  of  a  secondary  wavelet  to  the  point  of 
tangency  with  the  resulting  envelope,  bk,  for  instance,  is  nec- 
essarily perpendicular  to  the  envelope.  But  in  the  extraordi- 
nary light  this  is  not  necessarily  so,  because  the  secondary 
wavelets  are  ellipsoidal.  For  instance,  the  line  bl,  drawn  from 
b  to  the  point  where  the  ellipse  from  b  touches  the  extraor- 
dinary wavefront  sit,  is  not  perpendicular  to  the  latter,  and 
the  wavefront  advances,  not  perpendicular  to  itself  but  in  an 
inclined  direction.  "We  are  accustomed  to  speak  pf  two  veloci- 
ties for  the  extraordinary  light,  the  ray-velocity  and  the  wave- 
velocity,  which  are  in  the  ratio  of  bl  to  ok7.  For  the  ordinary 
light  ray-velocity  and  wave-velocity  are  the  same. 

108.  Special  cases  of  double  refraction. — If  the  incident 
light,  instead  of  striking  normally  upon  the  surface  of  the 
calcite  as  in  figure  109,  strikes  it 
obliquely,  the  two  wavefronts  in 
the  crystal  would  be  found  by 
the  method  of  Huyghens  in  a 
quite  analogous  way.  Figure  .110 
represents  such  a  case.  The  wave- 
front  XY  coming  from  the  air 
side  (right)  strikes  the  calcite 
surface  AB  first  at  Y.  While  it 
is  advancing  to  u  through  air, 
the  point  Y  sends  out  a  spherical 
wavefront  whose  radius  Yv  bears 
the  same  ratio  to  Xu  as  the  ve- 
locity of  the  ordinary  wraves  to 
the  velocity  in  air,  and  also  the 
corresponding  ellipsoidal  wavefront  as  shown.  At  later  instants 
the  points  b  and  a  take  up  the  role  of  secondary  centers  and 
send  out  spherical  and  ellipsoidal  wavefronts  of  correspondingly 
smaller  dimensions,  and  so  for  every  point  between  Y  and  u. 
The  common  tangent  plane  uv  to  all  the  spherical  wavelets  is 
the  ordinary  wavefront  in  the  crystal,  and  the  common  tangent 
plane  ut  to  all  the  ellipsoidal  wavelets  is  the  extraordinary. 


216 


LIGHT 


Here,  as  we  see,  the  0-waves  are  not  parallel  to  the  E-waves 
within  the  crystal. 

It  is  possible  to  cut  or  grind  a  piece  of  calcite  in  such  a 
way  that  the  optic  axis  is  parallel -to  the  surface,  although  the 
crystal  will  not  naturally  split  in  this  way.  The  part  of  figure 
111  to  the  left  of  AB  represents  calcite  cut  in  this  way,  AB 

being  the  outer  surface  and  the  optic 
axis  being  in  the  plane  of  the  paper 
parallel  to  AB.  The  Huyghens  con- 
struction is  shown  for  the  case  when 
a  train  of  plane  waves  XY  falls  at 
normal  incidence  upon  the  surface.  In 
this  case  there  is  no  lateral  displace- 
ment of  the  ordinary  waves,  and  in  a 
sense  one  may  say  there  is  no  refrac- 
tion, but  the  extraordinary  waves 
travel  faster  than  the  ordinary,  so  that 
if  they  both  emerge  into  the  air 
through  a  second  surface  parallel  to 
AB  the  E-waves  will  be  ahead  of  the 
0-waves  by  an  amount  which  depends 
Fi*ure  m  upon  the  thickness  of  the  layer  of  cal- 

cite. If  the  layer  is  thick  enough  so  that  one  wave  gets  a  quar- 
ter of  a  wavelength  ahead  of  the  other,  it  is  called  a  quart er- 
wave  plate,  if  thick  enough  so  that  one  gets  a  half-wavelength 
ahead  of  the  other  it  is  a  half-iuave  plate,  etc. 

109.  Tourmaline. — Certain  doubly-refracting  crystals  have 
the  peculiar  property  of  absorbing  one  of  the  rays  very  much 
more  than  the  other.  The  best-known  of  these  is  tourmaline, 
which  absorbs  the  ordinary  ray  so  strongly  that  two  or  three 
millimeters  of  the  crystal  practically  extinguish  it.  The  extra- 
ordinary ray  is  transmitted  with  little  absorption,  and  there- 
fore tourmaline  is  one  of  our  means  of  getting  a  beam  of  com- 
pletely polarized  light.  With  two  plates  of  tourmaline  a  very 
interesting  phenomenon  can  be  shown,  whose  explanation  is 
simple  in  terms  of  transverse  waves,  but  would  otherwise  prob- 
ably be  impossible.  If  the  plates  be  held  together  with  their 
principal  planes  parallel,  and  any  source  of  light  be  observed 
through  them,  the  light  can  be  plainly  seen,  though  less  than 


BIAXIAL  CRYSTALS  217 

half  as  bright  as  when  seen  directly,  for  of  course  half  the  light 
is  absorbed  as  ordinary  waves,  and  there  is  also  some  general 
absorption  and  loss  by  reflection.  But  if  one  of  the  plates  be 
turned  through  90°  about  the  beam  of  light  as  axis,  nothing  at 
all  can  be  seen  through  them.  The  light  transmitted  through 
the  first  plate  as  the  extraordinary  waves  has  its  plane  of 
polarization  in  such  a  direction  that  it  is  ordinary  light  for  the 
second  plate,  which  therefore  absorbs  it  completely.  In  this 
position  the  tourmalines  are  said  to  be  crossed. 

110.  Biaxial  crystals. — A  great  many  crystals  show  double 
refraction  in  a  way  different  from  calcite,  having  two  direc- 
tions instead  of  one  along  which  light  can 
be  transmitted  without  showing  double  im- 
ages. They  are  called  ~biaxial  crystals,  and 
both  rays  are  extraordinary,  following  laws 
which  are,  except  for  certain  special  direc-  Figure  nT 

tions,  more  complicated  than  the  simple  laws  of  refraction  that 
hold  for  non-crystalline  media.  The  complete  wavesurface  con- 
sists, not  in  a  sphere  and  an  ellipsoid,  but  in  a  very  complex 
surface  of  two  sheets.  Figure  112  is  a  perspective  view  of  a 
plaster  model  showing  one  fourth  of  the  complete  surface,  the 
equation  of  which  is 

y*  7?-      _ 

^"?  ~ 

where  r2  =  x2  +  y2  -|-  z2,  and  a,  b  and  c  are  certain  constants, 
having  different  values  for  different  crystals.  The  equation 
gives  the  wavefront  emitted  from  a  point,  at  a  definite  time,  say 
1  second  after  emission.  Perhaps  a  better  idea  of  the  shape  of 
the  surface  can  be  obtained  if,  after  studying  figure  112,  one 
considers  the  sections  made  by  the  three  coordinate  planes.  The 
section  with  the  YZ  plane  consists  of  a  circle  of  radius  a  and 
an  ellipse  of  semi-axes  b  and  c;  that  with  the  ZX  plane  is  a 
circle  of  radius  b  and  an  ellipse  of  semi-axes  c  and  a;  and 
that  with  the  XY  plane  is  a  circle  of  radius  c  and  an  ellipse 
of  semi-axes  a  and  b.  These  sections  are  shown  in  order  in 
figure  113,  which  is  drawn  on  the  assumption  that  a  is  greater 
than  b,  and  b  than  c.  If  it  should  happen  that  any  two  of 
these  quantities  are  equal,  the  crystal  would  become  uniaxial. 
If  all  three  are  equal,  it  would  not  be  doubly-refracting  at  all. 
This  is  the  case  for  rocksalt  and  some  other  crystals,  which  act, 


218 


LIGHT 


so  far  as  the  transmission  of  light  is  concerned,  like  glass  and 
other  isotropic  media. 


Figure   113 

111.  Poflarization  by  reflection. — In  1810  Malus  discov- 
ered quite  by  accident  that  the  light  reflected  from  glass  is  in 
general  partly  polarized.  In  looking  through  a  plate  of  tourma- 
line at  a  glass  window  from  which  the  direct  light  of  the  sun 
was  reflected,  he  found  that  when  the  tourmaline  was  turned  in 
a  certain  way  the  light  became  much  dimmer.  In  fact,  light  is 
polarized  to  a  greater  or  less  degree  when  it  is  reflected  from 
any  non-metallic  surface,  except  when  the  angle  of  incidence 
is  zero.  Observation,  by  means  of  a  tourmaline  or  any  equiva- 
lent instrument,  of  the  light  reflected  from  a  pond  of  water, 
a  slate  roof,  or  a  wet  cement  sidewalk,  will  show  always  a  cer- 
tain degree  of  polarization,  which  is  an  indication  that  the 
vibrations  in  a  certain  plane  are  stronger  than  in  a  plane  at 
right-angles.  This  statement  applies,  however,  only  to  light 
that  is  regularly,  not  diffusely  reflected.  Many  materials,  such 
as  paper,  which  diffuse  strongly,  also  reflect  regularly  to  some 
extent,  particularly  paper  which  has  a  strong  glaze.  The 
regularly  reflected  light  is  subject  to  polarization  by  the  act 

of  reflection,  but  the  diffusely  re- 
flected light  is  not.  In  fact,  the  de- 
gree of  polarization  of  the  reflected 
light  has  been  used  as  a  means  of 
grading  papers  in  regard  to  glaze. 

Figure  114  illustrates  the  polari- 
zation by  reflection  from  glass.    MN 
is  a  glass  slab,  T  a  plate  of  tourma- 
Figure  114  iin6j  in  sucn  position  that  its  princi- 

pal plane  is  perpendicular  to  the  plane  of  incidence  of  the  light 
on  MN,  i.  e.,  to  the  plane  of  the  paper  in  the  figure.     Under 


POLARIZATION  BY  REFLECTION  219 

these  circumstances,  the  reflected  light  passes  through  the, 
tourmaline,  but  if  the  latter  is  turned  about  the  reflected  ray 
CB  as  -axis,  till  its  principal  plane  is  parallel  to  the  plane  of 
incidence,  it  cuts  off  most  or  all  of  the  reflected  light,  de- 
pending upon  the  value  of  the  angle  of  incidence  ACD. 
There  is  a  certain  value  of  this  angle,  about  57°  for  the  common 
varieties  of  glass,  for  which  practically  all  the  light  is  cut  out 
by  the  tourmaline.  Since  this  light  is  transmitted  by  the  tour- 
maline when  the  plane  of  incidence  is  perpendicular  to  the 
principal  plane,  and  since  tourmaline  transmits  its  extraordi- 
nary light,  whose  plane  of  polarization  is  perpendicular  to  the 
principal  plane,  therefore  the  light  reflected  from  MN  is  polar- 
ized in  the  plane  of  incidence.  The  angle  of  incidence  for  which 
polarization  is  most  nearly  complete  is  called  the  angle  of 
polarization.  If  the  reflecting  surface  is  thoroughly  and  fresh- 
ly polished,  the  polarization  is  almost  perfect  at  the  angle  of 
polarization,  but  an  old  or  soiled  surface  polarizes  incompletely. 
T.he  mechanism  of  polarization  by  reflection  can  best  be 
explained  as  follows:  The  incident  light  coming  from  A,  being 
unpolarized,  has  its  vibrations  as  much  in  one  plane  through 
the  ray  AC  as  in  any  other,  but  every  vibration  can  be  re- 
solved into  a  component  vibration  in  the  plane  of  incidence 
and  one  at  right-angles.  _  Therefore,  we  may  regard  the  beam 
AC  as  composed  of  two  parts,  one  having  its  plane  of  polariza- 
tion in  the  plane  of  incidence,  the  other  at  right-angles  to  it. 
These  two  parts  are  unequally  reflected,  and  for  a  certain  angle 
of  incidence,  if  the  surface  is  fresh  and  clean,  the  second  is  not 
reflected  at  all.  Experiment  shows  that  if  A  represents  the 
amplitude  of  that  part  of  the  incident  light  which  is  polarized 
in  the  plane  of  incidence,  i  the  angle  of  incidence,  and  r  the 
angle  of  refraction,  the  amplitude  of  the  reflected  ray  to  which 
it  gives  rise  is 


sin  (i  -f  r) 
and  that  of  the  refracted  ray  is 

_       2  sin  r.  cos  i 

A     A  — — — ; T— 

sin  (i  -f-  r) 

But  if  B  represents  the  amplitude  of  that  part  of  the  incident 
light  whose  plane  of  polarization  is  perpendicular  to  the  plane 


220  LIGHT 

of  incidence,  B'  that  of  the  reflected  beam,  and  B"  that  of  the 
refracted  beam,  to  which  it  gives  rise, 

~,       -     tan  (i  —  r) 

JD    "=. 


B"  =  B  - 


tan  (i  +  r) 
2  sin  r  cos  i 


sin  (i  4-  r)  cos  (i  —  r) 

Theses  formulas  become  indeterminate  when  i  =  0,  but  hold 
good  for  any  angle  other  than  this.  None  of  the  numerators 
can  vanish  (except  when  i  =  0),  because  for  no  other  value  of 
i  can  i  —  r  =  0,  provided  there  is  any  change  in  the  medium 
at  all.  But  one  denominator,  that  in  the  expression  for  B', 
does  become  infinite,  if  i  -j-  r  =  90°.  Therefore,  when  the  angle 
of  incidence  becomes  large  enough — the  angle  of  refraction 
becoming  larger  along  with  it — so  that  the  two  of  them  together 
amount  to  90°,  B'  vanishes.  This  means  that  the  angle  of 
polarization  has  been  reached,  for  then  none  of  the  light  polar- 
ized perpendicular  to  the  plane  of  incidence  is  reflected  at  all, 
all  of  it  being  refracted.  The  only  reflected  light  is  then 
polarized  in  the  plane  of  incidence. 

Since  the  angle  of  polarization  is  the  angle  of  incidence 
when  the  reflected  light  is  all  polarized  in  the  plane  of  inci- 
dence, that  is  the  angle  such  that  i  +  r  =  90°,  we  can  derive 
a  simple  relation  between  this  angle  and  the  index  of  refrac- 
tion. For  then 

sin  r  =  cos  i 
and  since 

11  =  sin  i/sin  r 
we  have 

n  —  sin  i/cos  i  =  tan  i 

That  is,  the  index  of  refraction  is  the  tangent  of  the  angle  of 
polarization.  It  is  easy  to  prove,  by  simple  geometry,  that 
when  the  angle  of  incidence  is  the  polarizing  angle,  the  re- 
fracted ray  and  the  reflected  ray  make  an  angle  of  90°. 

When  the  plate  on  which  the  light  falls  at  the  polarizing 
angle  has  plane  and  parallel  sides,  as  indicated  in  figure  114, 
the  refracted  light  strikes  the  second  surface  at  what  is  the 
polarizing  angle  for  reflection  inside  the  plate.  Consequently, 
not  only  the  light  reflected  from  the  first  surface,  but  also  that 


POLARIZATION  BY  REFLECTION  221 

which  emerges  through  the  first  surface  after  any  odd  number 
of  reflections  inside  the  plate  is  polarized.  On  the  other  hand, 
the  transmitted  light,  although  it  shows  some  trace  of  polariza- 
tion, is  by  no  means  strongly  polarized.  For,  although  all  of 
the  light  whose  plane  of  polarization  is  perpendicular  to  the 
plane  of  incidence  is  refracted,  the  major  part  of  that  polarized 
in  the  plane  of  incidence  is  also,  so  that  the  transmitted  light 
has  only  an  excess  of  vibrations  in  one  plane. 

Since  a  glass  plate,  by  reflection,  gives  polarized  light,  a 
second  glass  plate  may  be  used  instead  of  a  tourmaline  to  de- 
tect the  polarization,  as  shown  in 
figure  115.  If  the  second  glass 
plate,  XY,  be  held  parallel  to  the 
first,  MN,  it  can  reflect  light  of 
the  same  kind  that  MN  reflects, 
for  their  planes  of  incidence  are 
paraUel.  But  if  XY  be  turned 
about  the  ray  CB  as  an  axis 
through  90°,  to  the  position  shown 
at  X'Y',  their  planes  of  incidence 
become  perpendicular,  and  the 
light  reflected  by  MN  cannot  be 
reflected  at  X'Y'.  Similarly  a  glass 
plate  can  be  used  to  test  any  beam  of  light  to  see  whether  it 
is  plane  polarized  or  not.  It  is  only  necessary  to  set  the  plate 
so  that  it  receives  the  beam  at  the  angle  of  polarization,  and 
then  turn  it  about  the  incident  beam  as  an  axis,  so  as  to  keep 
the  angle  of  incidence  always  equal  to  the  polarizing  angle.  If, 
for  any  position  of  the  plate,  the  reflected  light  vanishes,  the 
beam  in  question  is  polarized  in  a  plane  perpendicular  to  the 
plane  of  incidence. 

Problems. 

1.  What  must  be  the  angle  a,  figure  108,  in  order  that  the 
images  b  and  c  shall  be  twice  as  intense  as  the  images  a  and 
d! 

2.  Prove  that,  when  light  strikes    a    glass    plate    at  the 
polarizing  angle,  the  reflected  and  refracted  rays  are  at  right- 
angles. 


222  LIGHT 

3.  Find  the  angle  of  polarization  for  glass  whose  index  of 
refraction  is  1.71. 

4.  Show  that  light  that  has  gone  through  a  prism  spectro- 
scope must  be  partly  polarized.     In  what   plane  will  be  the 
maximum  polarization? 


CHAPTER  XIII. 

112.  Methods  of  polarizing  light.— 113.  The  Nicol  prism.— 114. 
Double-image  prisms. — 115.  .  Crossed  Nicols  and  crystal  plate. — 116. 
Elliptic  polarization. — 117.  Circular  polarization. — 118.  Rotation  of  the 
plane  of  polarization. — 119.  Magnetic  rotation. — 120.  The  rings-and- 
brushes  phenomenon. — 121.  The  nature  of  elliptic  and  circular  polar- 
ization. 

112.  Methods  of  polarizing  light. — For  many  purposes  in 
experimental  optics,  as  well  as  in  its  industrial  applications,  it 
is  desirable  to  obtain  a  strong  beam  of  plane  polarized  light. 
A  simple  block  of  calcite  does  not  answer  the  suppose,  for  it 
transmits  two  beams  with  mutually  perpendicular  planes  of 
vibration,  and  these  are  parallek  in  direction  of  propagation, 
with  only  a  lateral  displacement  which  is  too  small  to  separate 
them  completely.  Tourmaline  is  better,  since  the  ordinary 
beam  is  removed  by  absorption,  but  unfortunately  tourmaline 
exerts  a  fairly  strong  absorption  on  the  extraordinary  also, 
particularly  in  certain  wavelengths,  and  is  usually  strongly 
colored,  so  that  the  transmitted  beam  is  weak.  Reflection 
from  a  glass  plate  at  the  polarizing  angle  can  furnish  a  wide 
clear  beam  of  polarized  light,  free  from  absorption,  but  the 
following  calculation  will  show  that  this  device  utilizes  only  a 
small  portion  of  the  incident  light,  and  therefore  the  polarized 
beam  obtained  is  weak. 

If  we  take  the  index  of  refraction  of  the  glass  plate  as 
1.54,  and  remember  that  this  is  the. tangent  of  the  polarizing 
angle,  this  angle  is  found  to  be  57°.  With  an  angle  of  incidence 
of  57°  and  index  1.54,  the  angle  of  refraction  is  33°.  We  saw 
in  the  last  chapter  that  if  light  of  amplitude  A,  polarized  in 
the  plane  of  incidence,  strikes  the  surface,  the  amplitude  of  the 
reflected  light  is 


sin  (i  4-  r) 

If  we  substitute  i  =  57,  r  —  33,  we  get  A'  =  —  .407A,  that 
is  the  amplitude  of  the  reflected  ray  is  only  about  .4  that  of 
the  incident,  and  consequently  the  energy  of  the  reflected  light 
is  only  (.4)2  =  .16  that  of  the  incident.  That  is,  only  16  per- 

(223) 


224  LIGHT 

cent,  of  the  incident  energy  is  reflected,  84  percent  transmitted. 
All  this  is  on  the  supposition  that  the  incident  light  is  already 
polarized  in  the  plane  suitable  for  reflection.  In  reality,  we  are 
always  provided  with  a  completely  unpolarized  incident  beam, 
only  half  the  energy  of  which  is  polarized  in  the  plane  of  inci- 
dence, and  therefore  only  8  percent  of  the  incident  energy  is 
available  for  the  production  of  a  polarized  beam.  If  the  re- 
flecting plate  were  a  perfectly  efficient  polarizer,  the  reflected 
polarized  beam  would  contain  50  percent  of  the  incident  energy. 
It  is  true  that  the  efficiency  of  the  plate  is  raised  somewhat  it 
we  consider  the  light  reflected  from  the  second  surface,  but 
this  contribution  is  small,  and  the  presence  of  extra  images  by 
internal  reflection  is  usually  undesirable.  Glass  plates  used  for 
polarizing  by  reflection  are  usually  made  of  black  opaque  glass, 
so  as  to  avoid  these  extra  images. 

113.  The  Nicol  prism. — Figure  116  shows  an  end  and  a 
side  view  of  the  Nicol  prism,  the  best-known  device  for  obtain- 
ing polarized  light.  It  is  made  of  calcite  worked  in  such  a  way 
as  to  get  rid  of  the  ordinary  light  and  allow  the  extraordinary 
to  pass  through.  To  make  one,  a  rather  long  and  narrow  crystal 
of  calcite  is  cut  in  two  by  a  plane  through  the  obtuse  corners  A 
and  B.  The  two  halves,  after  being  ground  and  polished,  are 


Figure    116 

cemented  together  again  with  a  very  thin  layer  of  Canada 
balsam.  The  inclination  of  the  end  faces  is  also  slightly  altered 
from  the  natural  cleavage  surfaces.  Canada  balsam  is  a  muci- 
laginous substance  .which  has  an  index  of  refraction  less  than 
that  of  calcite  for  the  ordinary  light,  but  greater  than  the 
effective  index  of  the  calcite  for  extraordinary  light,  at  the 
angle  at  which  it  comes  in.  the  ordinary  use  of  the  prism.  For 
this  reason,  it  is  possible  for  the  ordinary  light  to  be  totally 
reflected,  if  the  angle  of  incidence  on  the  plane  AB  is  great 
enough,  while  the  extraordinary  cannot  be  totally  reflected,  no 
matter  what  this  angle  may  be.  The  slant  of  the  end  faces 
and  that  of  the  cut  AB  are  so  calculated  that  for  light  incident 


DOUBLE-IMAGE  PRISMS  225 

parallel  to  the  prism  axis,  or  for  a  few  degrees  each  side  of  it, 
the  ordinary  light  strikes  AB  at  greater  than  the  critical  angle. 
Consequently  the  ordinary  light  is  totally  reflected  off  to  the 
side  of  the  prism,  where  it  is  lost,  and  the  extraordinary  passes 
on  alone.  The  Nicol  prism  therefore  gives  a  clear  colorless 
beam  of  plane  polarized  light,  and  its  efficiency  is  high,  although 
of  course  the  emergent  light  is  to  some  extent  weakened  by 
reflection  where  the  beam  enters  and  leaves  the  prism. 

114.  Double-image  prisms. — There  are  also  several  devices 
in  which,  although  neither  the  ordinary  nor  the  extraordinary 
light  is  eliminated,  they  emerge,  not  with  parallel  rays  as  from 
a  simple  calcite  rhomb,  but  with  an  appreciable  angle  between 
them,  forming  a  diverging  pair  of  beams.  One  of  the  oldest 
is  the  Wollaston  prism,  figure  117.  It  is  made  of  two  wedges 
of  calcite,  ABC  and  ACD,  so  cut  that  in 
each  the  optic  axis  is  perpendicular  to  the 
entering  ray  of  light,  but  the  axis  in  one 
wedge  is  perpendicular  to  that  in  the  other. 
For  instance,  it  is  perpendicular  to  the 
plane  of  the  paper  in  ABC,  but  parallel  Figure  n? 

to  the  line  CD  in  ACD.  A  beam  of  unpolarized  light,  on 
entering  the  face  AB,  is  divided  into  an  ordinary  and  an 
extraordinary,  neither  of  which  is  bent  when  the  incidence  is 
normal,  because  the  optic  axis  is  parallel  to  the  face.  But  on 
striking  the  diagonal  face  AC,  the  ray  which  had  been  the 
ordinary  in  the  first  wedge  becomes  the  extraordinary '  in  the 
second,  and  vice  versa,  because  the  optic  axis  of  the  second  is 
perpendicular  to  that  of  the  first.  Therefore  one  beam  passes 
from  a  medium  where  it  has  less  velocity  to  one  where  it  has 
greater  and  is  therefore  bent  away  from  the  normal,  while  the 
other  passes  from  a  medium  where  it  has  greater  velocity  to 
one  where  it  has  less,  and  is  therefore  bent  toward  the  normal. 
Both  beams,  on  striking  the  air  at  the  surface  CD,  pass  into  a 
medium  of  greater  velocity  and  are  therefore  bent  away  from 
the  normal.  Thus  they  emerge  from  the  prism  with  a  con- 
siderable angle  of  divergence,  and  become  more  and  more  sep- 
arated as  they  recede  from  the  prism. 

The  Rochon  prism,  figure  118,  differs  from  the  Wollaston 
in  that  one  wedge  has  its  optic  axis  parallel  to  the  entering  ray, 


226  LIGHT- 

SO'  that  both  rays  travel  through  it  with  the  same  velocity.    One 
ray,  the  ordinary  for  the  second  wedge,  suffers  no  change  in 
speed   on   striking   the   diagonal  AC,  but 
the  other  undergoes  an  increase  in  speed 
-*— o      which  bends  it  away  from  the  normal.  The 
net  result  is  two  polarized  beams,  one  of 
which  goes    straight    through    the    prism 
Figure  118  without  any  bending  whatever,  while  the 

other  suffers  a  change  of  direction. 

Devices  like  the  Wollaston  and  Rochon  prisms,  which  give 
two  divergent  beams  of  polarized  light,  are  called  double-image 
prisms.  Prisms  are  now  made  which  differ  from  the  Rochon 
type  only  in  that  a  wedge  of  plain  glass  is  substituted  for  the 
wedge  whose  optic  axis  is  perpendicular  to  the  entering  face, 
and  they  work  very  well.  Double  image  prisms  are  useful  for 
many  optical  purposes,  but  where  only  a  single  beam  of  polar- 
ized light  is  desired  it  is  more  usual  to  employ  a  Nicol  prism. 

115.  Crossed  Nicols  and  crystal  plate. — Two  Nicol  prisms 
are  said  to  be  crossed  when  they  are  held  so  that  their  trans- 
mission planes  are  at  right-angles,  that  is,  so  that  the  polarized 
beam  transmitted  by  the  first  Nicol  is  refused  transmission  by 
the  second.     The  first  one  is  then  called  the  polarizer,  for  ob- 
vious reasons,  the  second  the  analyzer.   No  light,  of  course,  can 
be  seen  through  a  pair  of  Nicols  so  held,  but  if  a  thin  slip  of 
mica,  or  of  any  other  doubly-refracting  substance,  is  inserted 
between  the  polarizer  and  the  analyzer,  a  considerable  amount 
Of  light  will,  in  general,  pass  through  the  combination.     If  the 
slip  of  crystal  be  rotated  about  the  beam  of  light  as  an  axis, 
certain  positions  will  be  found  where    this    light    disappears, 
leaving  things  as  they  were  before  the  slip  was  inserted.     On 
the  other  hand,  if  the  slip  be  left  in  such  a  position  that  light 
passes  through,  and  the  analyzer  be  rotated,  the  brightness  of 
of  the  transmitted  light  may  show  fluctuations,  but  it  does  not 
vanish  completely  for  any  position  of  the  analyzer.     The  last 
fact  shows  that  the  action  of  the  crystal  slip  is  not  to  rotate 
the  plane  of  polarization,  but  either  to  depolarize  the  light  or 
to  change  it  to  a  new  kind  of  polarization  which  cannot  be 
shut  out  by  a  Nicol,  however  the  latter  be  held. 

116.  Elliptic  polarization. — It  is  in  fact  the  latter  alter- 
native which  holds  here,  that  is  the  experiment  introduces  us 


ELLIPTIC  POLARIZATION  227 

to  a  new  kind  of  polarization,  known  as  elliptic  polarization, 
which  can  be  explained  by  reference  to  figure  119.  Let  OA 
represent  in  direction  and  length 
the  amplitude  of  the  light  which  the 
polarizer  transmits.  Then,  since  the 
Nicols  are  crossed,  XY  is  the  direc- 
tion of  vibrations  which  alone  can 
be  transmitted  through  the  analyzer. 
Mica,  being  a  doubly-refracting  crys- 
tal, transmits  vibrations  in  two  mu- 
tually perpendicular  planes,  but  with 

different  velocities.  Suppose  that  OB  and  OC  are  these  two 
directions  for  transmission  through  the  mica.  The  amplitude 
OA  will  then  be  resolved  into  two  vibrations  of  amplitude 

OB  =  OA.  cos  a 

OC  =  OA.  sin  a 

These  two  are  in  the  same  phase  when  they  enter  the  mica,  but 
since  one  travels  faster  than  the  other  it  will  gain  in  phase  till 
they  both  emerge  again.  If  the  gain  in  phase  does  not  amount 
to  an  exact  multiple  of  ?r,  the  two  will,  on  emergence,  no  longer 
be  equivalent  to  the  linear  vibration  OA,  but  to  some  elliptical 
vibration,  the  elliptical  form  being  inscribable  within  the  rec- 
tangle of  dimensions  2 OB  X  20  C,  as  fully  explained  in  section 
02.  The  beam  of  light  coming  from  the  slip  of  mica  is  there- 
fore neither  unpolarized  nor  plane-polarized,  but  is  clearly  a 
very  particular  kind  of  vibration,  and  is  appropriately  known 
as  elliptically  polarized  light.  The  elliptical  vibration  of  figure 
]19,  which  is  equivalent  to  the  linear  vibrations  of  amplitude 
OB  and  OC  with  a  certain  phase-difference,  is  also  equivalent 
to  vibrations  of  amplitudes  OK  and  OL,  with  another  phase- 
difference.  The  vibration  OL  cannot  pass  through  the  analyzer, 
but  the  vibration  OK  can.  Consequently  light  is  seen  through 
the  combination  when .  the  mica  slip  is  inserted.  If  the  plane 
of  transmission  of  the  analyzer,  XY  is  rotated,  the  amplitude  of 
the  transmitted  component  will  vary,  its  greatest  and  least 
values  being  the  major  and  minor  semi-axes  of  the  ellipse,  but 
it  will  never  vanish.  On  the  other  hand,  suppose  that  the  ana- 
lyzer is  held  in  the  position  shown  in  the  figure,  but  the  crystal 
slip  is  rotated,  so  that  the  angle  a  changes.  "When  a  is  zero 


228  LIGHT 

or  180°,  OB  is  parallel  to  OA  and  equal  to  it  in  absolute 
amount,  while  OC  is  zero.  Then  only  a  single  beam  goes 
through  the  mica  and  it  is  in  such  a  direction  as  to  fail  to  pass 
through  the  analyzer.  If  a  is  90°  or  270°,  OC  is  parallel  to 
OA  and  equal  to  it  in  absolute  amount,  while  OB  is  zero.  Again 
only  a  single  beam  goes  through  the  mica  and  it  is  in  such  a 
direction  as  to  fail  to  pass  the  analyzer. 

117.  Circular  polarization. — If,  in  figure  119,  a  =  45°, 
OB  and  OC  are  equal.  If,  in  addition  the  difference  in  phase 
of  the  two  beams  through  the  mica  is  7r/2,  or  90°,  the  ellipse 
becomes  a  circle,  and  we  have  emerging  from  the  mica  what 
we  call  circMlarly -polarized  light.  Evidently  circularly  polar- 
ized light  may  be  right-handed  or  left-handed,  according  to 
which  of  the  two  component  beams  passes  through  the  mica 
with  the  greater  velocity.  (The  same  is  true  of  elliptically- 
polarized  light.)  Now  a  difference  of  phase  of  -Tr/2  evidently 
means  that  one  of  the  beams  passing  through  the  mica  gains  a 
quarter  of  a  wavelength  over  the  other,  and  as  we  saw  in  the 
preceding  chapter  a  plate  thick  enough  for  one  component  to- 
gain  just  a  quarter  of  a  wavelength  over  the  other  is  what  is 
known  as  a  quarter-wave  plate.  Accordingly,  we  have  the 
following  rule  for  the  production  of  circularly  polarized  light: 
Place  a  quarter-wave  plate  in  the  path  of  a  beam  of  plane- 
polarized  light,  so  that  its  two  planes  of  transmission  make  an 
angle  of  45°  with  the  plane  of  polarization  of  the  incident  light. 
The  transmitted  light  is  then  circularly  polarized.  Quarter- 
wave  plates  are  usually  made  of  mica  because  it  splits  into 
layers  so  readily.  It  is  peeled  down,  a  thin  layer  at  a  time, 
to  the  required  thickness. 

A  quarter-wave  plate  made  of  calcite  would  have  to  be 
exceedingly  thin,  on  account  of  the  great  difference,  in  this 
crystal,  between  the  ordinary  and  the  extraordinary  velocities. 
It  is  true,  that,  theoretically,  a  plate  SQ  thick  that  one  wave 
gains  over  the  other  a  quarter  of  a  wavelength  plus  any  whole 
number  of  wavelengths  would  act  as  a  simple  quarter-wave 
plate,  and  it  would  so  act  in  practice  if  the  light  used  were 
monochromatic.  But  doubly-refracting  substances,  like  iso tropic 
materials,  are  subject  to  dispersion,  or  differences  in  velocity 


COLORS  FROM  CRYSTAL  PLATES       229 

for  different  wavelengths,  and  the  dispersion  for  the  ordinary 
and  for  the  extraordinary  light  is  not  the  same.  For  example, 
a  plate  thick  enough  for  the  extraordinary  ray  to  gain  10% 
wavelengths  over  the  ordinary  in  the  yellow  might  show  a 
difference  of  10%  in  the  red,  9%  in  the  green,  9  in  the  blue, 
and  8y2  in  the  violet,  these  figures  being  merely  illustrative, 
and  not  actual  statements  of  what  any  particular  plate  would 
do.  Under  these  circumstances,  certain  colors  would  be  cut  out 
completely,  all  those  in  fact  for  which  the  gain  was.  an  exact 
whole  number  of  wavelengths.  For  an  examination  of  the 
figure  will  show  that  for  these  colors  the  rays  of  amplitude 
represented  by  OB  and  OC  would  emerge  from  the  plate  in  the 
same  phase,  and  would  therefore  recombine  to  produce  the 
original  plane  polarized  light  of  amplitude  OA.  Of  the  other 
colors,  some  would  be  represented  by  right-handed,  some  by 
left-handed  circularly  polarized  light,  and  others  by  elliptically 
polarized  light  of  various  configurations  of  the  ellipse.  Con- 
sequently, of  the  light  passing  through  the  analyzer,  parts  of 
the  spectrum  would  be  completely  eliminated,  other  parts  much 
weakened,  arid  still  other  parts  quite  strong.  The  result  would 
be  brilliant  color  effects,  which  would  change  when  either  the 
analyzer  or  the  crystal  plane  was  rotated.  On  the  other  hand, 
if  the  gain  of  the  extraordinary  over  the  ordinary  ray  were 
only  %  wavelength  for  any  particular  color,  it  would  not  differ 
a  great  deal  from  that  for  any  visible  wavelength,  and  there 
would  hardly  be  a  suggestion  of  color  in  whatever  light  got 
through  the  analyzer. 

The  phenomena  explained  above  afford  a  very  convenient 
and  delicate  test  for  double  refraction.  It  is  only  necessary  to 
set  up  a  pair  of  crossed  Nicols  and  insert  between  them  a  slice 
of  the  material  to  be  investigated.  If,  when  this  is  rotated 
into  various  positions,  no  light  passes  through  the  analyzer,  the 
material  is  free  from  double  refraction.  Glass  is  of  this  char- 
acter if  carefully  annealed  and  not  strained,  but  a  piece  of 
commercial  glass-ware,  if  experimented  with  between  crossed 
Nicols,  is  almost  sure  to  show  streaks  of  light  which  indicate  an 
irregular  double  refraction,  and  even  well  annealed  glass,  if 
bent  between  the  fingers,  or  otherwise  slightly  strained,  shows 
the  same  effect,  though  the  degree  of  double  refraction  is  far 
too  small  to  show  double  images  to  the  unaided  eye. 


230  LIGHT 

118.  Rotation  of  the  plane  of  polarization. — Quartz  is  a 
doubly-refracting  crystal,  and  shows  phenomena  similar  to 
those  of  calcite,  though  to  a  far  less  degree,  the  difference  in 
the  velocities  of  the  two  rays  being  always  small.  But  quartz 
shows  in  addition  a  very  remarkable  property  which  calcite 
and  most  other  doubly-refracting  crystals  do  not  share,  what 
is  called  optical  rotation.  When  a  beam  of  light  is  passed 
through  quartz  along  the  optic  axis,  just  the  condition  under 
which  double  refraction  does  not  occur,  the  plane  of  polariza- 
tion is  turned  without  changing  the  direction  of  the  ray,  and 
the  number  of  degrees  of  turning  is  directly  proportional  to 
the  thickness  of  quartz  passed  through.  Some  samples  of 
quartz  twist  the  plane  to  the  right,  others  to  the  left,  and  in 
all  samples  the  amount  of  turning  is  very  different  for  different 
wavelengths. 

If  a  layer  of  quartz  crystal,  cut  so  that  the  optic  axis  is 
perpendicular  to  the  faces,  is  inserted  between  a  pair  of  crossed 
Nicols,  light  reappears,  just  as  when  a  slip  of  mica  is  so  in- 
serted, but  it  is  very  easy  to  distinguish  between  the  two 
effects.  With  the  slip  of  mica,  as  we  have  already  seen,  a 
rotation  of  the  slip  shows  certain  positions  where  the  light 
vanishes  when  the  Nicols  remain  crossed,  but  a  rotation  of  the 
quartz  has  no  effect,  so  long  as  the  optic  axis  remains  parallel 
to  the  beam.  On  the  other  hand,  with  the  mica  fixed  in  position 
so  as  to  show  light  through  the  crossed  Nicols,  a  rotation  of  the 
analyzer  may  cause  fluctuations  in  the  transmitted  light,  but 
does  not  quench  it  entirely,  while  with  the  quartz  a  rotation  of 
the  analyzer  will  enable  the  light  to  be  completely  cut  out. 
These  facts  show  that  the  effect  of  the  quartz  is  not  to  produce 
elliptically  polarized  light  like  the  mica,  but  to  leave  the  light 
plane  polarized  but  with  a  changed  azimuth  of  the^plane  of 
polarization.  Since  quartz  is  itself  a  doubly-refracting  crystal, 
it  too  would  produce  elliptical  polarization  if  the  light  went 
through  it  perpendicular  to  the  optic  axis,  and  when  the  ray 
is  inclined  to  the  optic  axis  the  effect  is  a  combination  of  double 
refraction  and  rotation  which  is  complicated  and  difficult  to 
describe. 

The  power  to  rotate  the  plane  of  polarization  is  shown  by 
a  number  of  other  crystals  besides  quartz,  and  also  by  certain 
solutions,  notably  the  sugars,  and  it  forms  the  basis  of  an 


OPTICAL  ROTATION 


231 


elaborate  technical  method  of  analyzing  and  testing  commer- 
cial sugars.  Incidentally,  since  in  solutions  there  can  be  no 
question  of  a  regular  arrangement  of  molecules  such  as  occurs 
in  crystals,  it  seems  certain  that  solutions  must  owe  their  rotat- 
ing power  to  something  in  the  interior  structure  of  the  molecule. 

Fresnel  has  offered  a  very  ingenious  kinematical  explana- 
tion of  optical  rotation.  Taking  a  suggestion  from  the  fact 
that  in  ordinary  double  refraction  linear  vibrations  in  two 
perpendicular  planes  are  transmitted  through  the  crystal  with 
different  velocities,  he  made  the  supposition  that  along  the 
optic  axis  of  quartz  circular  vibrations  in  two  directions,  viz., 
right-handed  and  left-handed,  are  transmitted  with  different 
velocities,  and  from  this  supposition  the  rotation  of  the  plane 
of  pZflwe-polarized  light  follows  as  a  logical  deduction.  We 
have  shown  in  section  92  that  a  circular  vibration  is  equivalent 
to  two  linear  vibrations,  at  right-angles  in  direction  and  with 
a  phase-difference  of  w/2,  and  we  can  now  show  that  conversely 
a  linear  vibration  is  equivalent  to  a  right-handed  and  a  left- 
handed  circular  vibration,  each  having  half  the  amplitude  of 
the  linear  vibration.  This  can.  be  done  analytically,  by  the 
use  of  equations,  but  the  following  graphical  method  is  perhaps 
better. 

A  right-handed  circular  motion  can  be  represented  by  a 
vector,  whose  length  is  equal  to  the  radius  of  the  circle,  and 
which  swings  at  a  uniform  rate  about 
the  origin,  like  the  vector  OA  in  figure 
120,  that  is,  the  end  of  the  vector  al- 
ways gives  the  position  of  the  body 
undergoing  the  circular  motion.  A 
left-handed  circular  motion  would  be 
similarly  represented  by  a  vector 
swinging  in  the  opposite  direction. 
The  rule  for  compounding  two  vectors 
is  to  place  the  origin  of  one  at  the  ter- 
minus of  the  other.  Therefore,  to  com- 
pound a  left-handed  with  a  right- 
handed  circular  motion,  we  take  the 
terminus,  A,  of  the  latter  for  the  center  about  which  the  former, 
AB,  turns.  Now  if  OA  swings  at  a  uniform  rate  to  the  right 
while  AB  swings  at  a  uniform  rate  to  the  left  from  the  end  of 


Figure    120 


232  LIGHT 

OA,  the  point  B  will  describe  the  path  of  a  point  whose  motion 
is  a  combination  of  the  two  circular  motions.  If  OA  and  AB  are 
equal  in  length,  and  if  they  both  rotate  at  the  same  rate,  this 
path  is  the  straight  line  CD,  as  can  be  easily  proved  by  simple 
geometry.  It  is  also  easy  to  prove  that  the  motion  is  simple- 
harmonic,  with  amplitude  twice  the  radius  of  either  circle. 
Any  plane-polarized  beam  of  light  can  therefore  be  regarded 
as  composed  of  two  circularly-polarized  components  of  the  same 
period  but  half  the  amplitude,  one  right-handed,  the  other  left- 
handed.  According  to  Fresnel's  hypothesis,  these,  on  entering 
the  quartz  along  the  optic  axis,  will  travel  through  it  with 
different  velocities.  On  emerging  into  the  air  again,  one  of 
these  components  will  have  gained  on  the  other  in  phase,  be- 
cause of  its  greater  velocity,  and  this  gain  in  phase  causes  the 
plane-polarized  beam  to  which  the  two  on  emergence  are  equiv- 
alent to  be  turned  somewhat  from  the  plane  of  polarization  of 
the  light  as  it  entered  the  quartz.  Irii  order  to  explain  thisr 
suppose  that  the  right-handed  vibration  in  the  figure  is  % 
revolution  ahead  of  the  left-handed.  Then,  when  the  former 
had  turned  90°  from  the  vertical,  into  the  position  OA',  the 
latter  would  be  just  passing  through  the  vertical  position,  as 
A'B'.  Consequently,  the  path  of  the  tracing  point  would  be 
along  the  diagonal  OC'  instead  of  OC,  and  the  plane  of  the 
vibration  would  be  rotated  through  45°.  Evidently,  the  angle 
through  which  the  plane  of  polarization  is  rotated  is  half  the 
gain  in  phase  of  one  component  over  the  other,  and  therefore 
it  will  be  proportional  to  the  length  of  path  traversed  in  the 
quartz. 

Fresnel  subjected  his  theory  to  a  further  interesting  test. 
If  the  two  kinds  of  circularly  polarized  light  travel  with  differ- 
ent velocities  within  the  quartz,  they  will  have  different  indices 
of  refraction.  Now  suppose  that  plane  polarized  light  be  sent 
through  a  prism  cut  out  of  quartz  in  such  a  way  that  the  optic 
axis  is  parallel  to  the  base  of  the  prism.  The  two  circularly- 
polarized  components  to  which  the  plane-polarized  beam  is 
equivalent,  having  different  indices  of  refraction,  should  sepa- 
rate, just  as  two  different  wavelengths  separate  in  going  through 
a  glass  prism,  emerge  with  different  directions,  and  be  brought 
to  a  focus  at  different  points.  This  actually  proves  to  be  the 
case.  Light  of  a  single  wavelength,  when  sent  through  a  spec- 


MAGNETIC  ROTATION  233 

troscope  with  such  a  prism  gives  two  spectral  lines  instead  of 
one,  one  line  composed  of  right-handed,  the  other  of  left-handed 
circularly  polarized  light,  but  both  having  the  original  wave- 
length. When  Fresnel  tried  this  experiment,  he  actually  used 
a  row  of  prisms  instead  of  one,  a  prism  of  right-handed  quartz 
followed  by  one  of  left-handed  quartz  with  its  base  in  the 
opposite  direction,  this  in  turn  followed  by  a  prism  like  the 
first,  and  so  on.  By  this  arrangement,  the  effect  of  a  single 
prism  in  separating  the  two  beams  is  greatly  increased,  but 
such  an  elaborate  arrangement  is  not  necessary.  A  single  60° 
prism  of  ordinary  size  is  sufficient  to  show  the  effect,  and  in- 
deed this  doubling  of  the  spectrum  lines  is  so  troublesome  that 
whenever  quartz  prisms  are  used  for  studying  ultraviolet  light, 
they  must  be  made  double,  half  a  prism  being  of  right-handed, 
half  of  left-handed  quartz,  the  two  halves  having  their  bases 
in  line  and  their  apexes  together.  By  this  contrivance  the 
effect  is  neutralized,  one  prism  undoing  the  rotation  produced 
by  the  other,  and  single  spectral  lines  result. 

It  should  be  noted  that  the  doubling  of  spectral  lines  by 
a  quartz  prism  can  be  explained  without  making  use  of  Fres- 
nel's  hypothesis,  as  a  simple  result  of  rotation  and  diffraction, 
but  there  can  be  no  doubt  that  right-handed  and  left-handed 
light  does  pass  along  the  optic  axis  of  quartz  with  different 
velocities. 

119.  Magnetic  rotation. — Michael  Faraday  found  that 
when  certain  substances  are  placed  within  a  strong  magnetic 
field  and  plane-polarized  light  is  sent  through  them  along  the 
magnetic  lines  of  force,  the  plane  of  polarization  is  rotated 
just  as  it  is  in  a  sugar  solution,  or  in  quartz  along  the  optic 
axis.  This  discovery  was  the  first  intimation  of  any  connection 
between  optical  and  electromagnetic  phenomena,  and  is  one  of 
the  facts  that  stimulated  the  electromagnetic  theory  of  light, 
which  we  are  to  take  up  later.  A  similar  phenomenon  was 
discovered  by  Kerr.  He  found  that  when  a  beam  of  light, 
polarized  in  or  at  right-angles  to  the  plane  of  incidence,  is 
reflected  from  the  polished  pole  piece  of  a  strong  magnet,  the 
plane  of  polarization  is  slightly  rotated  in  the  reflected  light. 

There  is  one  important  difference  between  the  rotation  in 
quartz  and  other  crystals  or  solutions,  and  the  rotation  due  to 
magnetic  action  discovered  by  Faraday.  In  the  former,  if  the 


234  LIGHT 

rotation  is  right-handed,  for  instance,  going  in  one  direction,  it 
will  also  be  right-handed  going  in  the  opposite  direction.  Stat- 
ing the  matter  in  another  form,  if  a  beam  be  sent  along  the 
optic  axis  of  a  piece  of  right-handed  quartz,  so  that  the  plane 
of  polarization  is  turned  to  the  right,  and  then  reflected  .back 
over  its  path  by  means  of  a  mirror,  the  plane  will  again  be 
rotated  to  the  right,  bringing  it  back  exactly  to  the  azimuth 
which  it  originally  had,  for  a  right-handed  rotation  with  re- 
versed path  undoes  the  effect  of  the  original  right-handed 
rotation.  On  the  other  hand,  in  the  case  of  magnetic  rotation, 
the  turning  is  to  the  right  when  the  light  goes  out  in  the  direc- 
tion of  the  magnetic  lines  of  force,  but  to  the  left  when  it  goes 
against  the  latter,  so  that  a  beam  sent  out  along  the  lines  of 
force  and  reflected  back  suffers  twice  as  much  rotation  as  if  it 
traversed  the  distance  only  once.  In  magnetic  rotation,  the 
direction  of  rotation  depends  upon  the  direction  of  the  lines  of 
force  of  the  magnetic  field,  but  in  rotation  produced  by  crys- 
tals and  solutions  it  depends  upon  the  direction  of  the  ray. 

120.  The  rings-and-brushes  phenomenon. — Some  very  re- 
markable and  beautiful  effects  are  produced  when  a  beam  of 
plane-polarized  light  is  passed  through  a  thin  layer  of  doubly- 
refracting  crystal  cut  with  its  faces  perpendicular  to  the  optic 
axis,  and  the  light  is  received  through  an  analyzing  Nicol 
prism  and  a  telescope.  Thus,  let  AB,  figure  121,  represent  a 
slip  of  some  uniaxial  crystal  which  is  free  from  the  rotating 

power  of  quartz.  CD,  perpendicular 
to  the  faces  of  the  slip,  shows  the 
direction  of  the  optic  axis,  and  the 
polarized  light  comes  through  from 
below,  in  a  cone  whose  axis  is  paral-, 
lei  to  this  line.  The  axial  ray  CD 
obviously  suffers  no  double  refrac- 
tion, but  a  ray  ED,  inclined  to  the 
optic  axis,  is  resolved  into  an  ordi- 
nary and  an  extraordinary  ray,  and  these  emerge  from  the 
crystal  parallel  to  one  another  and  with  a  phase-differ- 
ence, unless  the  plane  of  polarization  of  the  incident  light 
happens  to  be  either  parallel  or  perpendicular  to  the  plane 
through  ED  and  the  optic  axis,  the  plane  of  the  paper 


RINGS^AND  BRUSHES 


235 


for  this  case.  The  phase-difference  between  ordinary  and  extra- 
ordinary will  depend  upon  the  inclination  and  length  of  the 
path  within  the  crystal,  for  instance  it  will  evidently  be  greater 
for  the  ray  FD  than  for  ED,  since  FD  is  more  inclined  to  the 
axis  and  has  also  a  longer  path  within  the  crystal. 

The  square  area  in  figure  122  represents  the  crystal  slip 
as  seen  from  above,  the  optic  axis  now  being  perpendicular  to 
the  paper.  Let  0  be  the  point  of  emergence  of  the  axial  ray 
Cd  of  figure  121,  and  AB  the  plane  of  polarization  of  the  inci- 
dent light.  Referring  back  to  figure  121,  it  is  clear  that  the 
phase-difference  that  exists  between  the  components  of  the  ray 


c— 


,r 


•-<^    %NJ/ 
-«?»     V  \ 


/  /*.  *. 


-i—  i— -i— -i—  -,t-'-~  t— -i — t— t- 


Figure   122 

ED  will  be  the  same  as  for  any  other  ray  such  as  E'D  which 
emerges  from  the  crystal  at  the  same  distance  from  the  point 
0  of  figure  122.  Therefore,  if  in  the  latter  figure  we  draw 
circles  about  0,  the  phase-difference  between  the  ordinary  and 
the  extraordinary  rays  on  emergence  will  be  the  same  for  all 
points  on  any  one  circle,  but  different  for  points  on  different 
circles.  Circles  are  drawn  for  which  this  phase  difference  is  TT, 
2-7T,  STT,  4rr,  respectively. 

Consider  first  the  light  that  comes  through  at  points  along 
the  line  AB.     For  these  points  the  principal  plane  is  the  same 


236  LIGHT 

as  the  plane  of  polarization,  therefore  there  is  no  extraordinary 
ray  and  all  the  light  passes  through  as  ordinary  light.  Next 
take  points  along  CD.  For  these  the  principal  plane  is  per- 
pendicular to  the  plane  of  polarization  of  the  incident  light, 
and  therefore  there  is  no  ordinary  ray,  and  all  the  light  comes 
through  as  extraordinary  ray  with  its  plane  of  polarization  un- 
altered. Points  along  AB  and  CD  would  then  be  represented 
by  plane-polarized  light  with  the  same  plane,  as  indicated  by 
the  arrows  along  these  two  lines.  Now  take  points  along  EF 
or  GH.  Here  the  principal  plane  is  at  45°  with  the  plane  of 
polarization  of  the  incident  light,  there  will  be  both  an  ordinary 
and  an  extraordinary  ray,  and  their  amplitudes  will  be  equal, 
but  the  phase-difference  between  them  will  depend,  as  we  have 
seen,  upon  the  distance  of  the  point  from  the  center.  At  O 
itself,  and  also  where  EF  and  GH  cross  the  circles  marked  2?r, 
47r,  etc.,  the  two  rays  are  equivalent  to  the  original  polarized 
incident  ray,  for  a  phase-difference  of  2tr,  or  any  even  multiple 
of  TT,  is  equivalent  to  no  phase-difference  at  all.  Consequently 
we  mark  arrows  at  these  points  parallel  to  the  plane  of  polari- 
zation of  the  incident  light  and  to  the  arrows  along  AB  and 
CD.  But  where  EF  and  GH  cross  the  circles  marked  TT,  STT, 
etc.,  (any  odd  multiple  of  TT  is  equivalent  to  TT  when  we  are 
speaking  of  phase-differences)  the  two  rays  are  equivalent  to 
a  ray  polarized  at  right-angles  to  the  incident  light,  as  can  be 
seen  by  comparing  the  diagram  of  figure  94  for  ft  =  TT,  with 
that  for  ft  =  0  or  /?  = .  2ir.  These  points  are  marked  with 
arrows  at  right-angles  to  those  along  AB  and  CD.  The  reader 
should  now  be  able  to  foresee  what  happens  at  points  where 
the  circles  are. cut  by  lines  making  any  other  angle,  say  22.5°, 
with.  AB  or  CD.  At  all  these  points  there  will  be  both  an 
ordinary  and  an  extraordinary  ray,  of  unequal  amplitudes. 
At  the  circles  of  phase-difference  27r,  or  4?r,  the  original  plane 
of  polarization  is  reproduced,  but  at  the  TT  or  BTT  circles  the 
result  will  be  plane  polarized  light,  but  with  an  inclined  direc- 
tion. Arrows  are  marked  only  in  the  upper  left-hand  quad- 
rant of  the  figure  to  indicate  the  azimuth  of  the  plane-polar- 
ized light  at  points  on  these  circles. 

Other  circles  might  be  drawn,  for  which  the  phase-differ- 
ence is  7T/2,  37T/2,  57T/2,  etc.,  and  it  is  not  difficult  to  show  that 
at  various  points  on  these  circles  the  resulting  light  is  circularly 


NATURE  OF  POLARIZED  LIGHT  237 

or  elliptically  polarized,  except  where  they  are  crossed  by  the 
lines  AB  and  CD,  where  the  ellipse  becomes  a  straight  line. 

We  have  shown  that  the  light  coming  through  at  various 
places  on  the  plate  would  have  diverse  states  of  polarization, 
but  nothing  of  all  this  would  show  to  the  eye  without  the  use 
of  a  second  Nicol  prism  or  other  analyzer,  for  the  eye  cannot 
detect  polarization.  If  an  analyzer  is  inserted,  parallel  to  the 
polarizer  which  furnishes  the  incident  light,  brightness  will 
show  at  all  those  points  of  the  diagram  marked  with  arrows 
like  those  along  AB  or  CD,  complete  darkness  at  points  marked 
with  arrows  at  right-angles  to  these,  and  partial  illumination 
at  points  where  the  polarization  is  elliptical  or  circular,  or  plane 
but  inclined  to  the  line  AB.  There  will  be  a  bright  cross  AB 
and  CD,  cutting  across  a  series  of  bright  and  dark  rings.  ^  If 
the  analyzer  is  crossed  with  the  polarizer,  the  pattern  will  be 
exactly  reversed,  consisting  of  a  black  cross  cutting  across  a 
ring-system.  This  is  the  phenomenon  of  "rings  and  brushes," 
the  name  being  suggested  by  the  appearance. 

121.  The  nature  of  elliptic  and  circular  polarization. — • 
Students  often  find  difficulty  in  forming  a  satisfactory  physical 
conception  of  circularly-polarized  waves,  or  of  unpolarized 
waves,  although  plane-polarization  may  offer  no  particular 
difficulties.  Probably  this  may  be  overcome  most  easily  by  con- 
sidering simple  mechanical  waves  in  an  elastic  jelly,  as  was 
done  in  chapter  III  to  explain  plane  wavefronts.  Figure  17  of 
that  chapter  represents  a  block  of  jelly  with  a  stiff  board  at- 
tached to  it  so  that  any  movement  of  the  board  in  its  own  plane 
sends  a  train  of  transverse  plane  waves  through  the  jelly.  It 
is  clear  enough  that  a  movement  of  the  board  back  and  forward 
in  the  direction  AB,  or  along  any  line  in  the  plane  of  the  board 
inclined  to  AB  at  any  angle,  would  cause  the  waves  produced 
to  be  plane-polarized.  In  order  to  make  circularly-polarized 
waves,  the  motion  of  the  board  would  have  to  be  circular  in 
its  own  plane,  but  the  motion  must  be  one  of  translation,  not 
of  rotation.  The  board  must  not  turn  about  a  fixed  axis,  like- 
a  wheel  on  its  shaft,  but  must  move  so  that  its  edges  remain 
parallel  to  their  initial  directions  and  so  that  every  point  in 
the  board  moves  in  a  circle  of  the  same  diameter.  Then  one 
plane  after  another  in  the  jelly  would  take  up  the  motion,  and 


238  LIGHT 

circularly-polarized  wavefronts  would  advance  through  it.  If 
the  path  of  every  point  in  the  board  were  an  ellipse,  the  same 
would  be  true  of  each  point  in  the  jelly  when  the  wavefront 
reached  it,  and  the  waves  would  be  elliptically  polarized.  In 
order  to  produce  unpolarized  waves,  the  Aboard  must  move  so 
that,  although  its  edges  keep  parallel  to  themselves,  the  path 
of  each  point  in  it  would  be  a  very  irregular  and  random  sort 
of  curve.  Each  point  in  the  jelly  would  in  its  turn  go  through 
the  same  path,  and  no  particular  direction  of  vibration  would 
predominate.  • 

Whether  light  vibrations  actually  consist  of  real  mechani- 
cal displacements  in  the  ether,  like  the  mechanical  displace- 
ments in  the  jelly,  is  a  question  which  up  to  the  present  we 
have  ignored.  But  it  seems  certain  at  any  rate  that  whatever 
may  be  the  character  of  the  ether-disturbances  which  produce 
light,  they  must  be  of  the  nature  of  vectors,  since  the  phe- 
nomena of  polarization  clearly  indicate  that  they  have  direc- 
tion. Therefore  it  is  convenient  and  permissible  to  represent 
them  as  mechanical  displacements,  remembering  that  they  may 
prove  to  be  something  else. 

Problems. 

1.  A  device  that  has  been  used  for  producing  polarized 
light  is  a  pile  of  plates  set  so  that  the  light  passing  through 
strikes  each  at  the  polarizing  angle.     If  each  reflection   (two 
for  every  plate)  reduces  by  16%  the  energy  of  that  part  of  the 
beam  incident  upon  it  which  is  polarized  in  the  plane  of  inci- 
dence, what  would    be    the    total    percentage  reduction  by  10 
plates $ 

2.  Light  passes  through  one  Nicol  prism  and  then  through 
another  whose  plane  of  transmission  makes  an  angle  a  with 
that  of  the  first.    Neglecting  losses  by  reflection  from  the  end 
faces,  what  must  be  the  angle  a  in  order  that  the  second  Nicol 
cut  down  the  intensity  to  %?.    to  %?    to  %? 

3.  Suppose  a  beam  of  light,  whose  origin  is  unknown,  is 
passed  into   a  room.     What  instruments  would  be  used,  and 
how  would  one  proceed,  to    find    whether    it    is    unpolarized, 
plane-polarized,   circularly  polarized,   elliptically   polarized,   or 
partly  plane-polarized  ? 


PROBLEMS  239 

4.  Mica  transmits,  perpendicularly  to  its  natural  cleavage 
planes,  two  beams  whose 'indices  are  1.5609  and  1.5941,  their 
planes  of  polarization  being  of  course  mutually  perpendicular. 
These  indices  are  for  light  of  wavelength  .00005893.  What  will 
be  the  thickness  of  a  quarter-wave  plate  of  mica  for  this  wave- 
length? 


CHAPTER  XIV. 

122.  Plane  of  polarization  and  plane  of  vibration.— 123.  Elastic- 
solid  theories. — 124.  Electromagnetic  theory. — 125.  Direction  of  the 
vibrations. — 126.  Fundamental  electromagnetic  laws. — 127.  Faraday's 
displacement-currents. — 128.  Maxwell's  assumptions. — 129.  Hertz's  ex- 
periments.— 130.  Propagation  of  electromagnetic  waves. — 131.  Velocity 
of  the  waves. — 132.  Refractive  index  and  dielectric  constant. 

122.  Plane  of  polarization  and  plane  of  vibration. — It  has 

been  shown  that,  in  the  ordinary  and  the  extraordinary  waves 
produced  by  double  refraction,  the  directions  of  the  vibrations 
—or,  more  precisely,  the  directions  of  the  light-disturbance 
vectors — are  at  right-angles  to  one  another.  But  no  evidence 
was  shown  to  indicate  whether  this  vector  for  the  ordinary  or 
for  the  extraordinary  lies  in  the  principal  plane.  The  difficulty 
was  temporarily  avoided  by  agreeing  to  call  the  principal  plane 
the  ' '  plane  of  polarization ' '  of  the  ordinary  waves ;  and  con- 
sistently therewith,  the  plane  of  polarization  for  the  extraordi- 
nary waves  must  be  perpendicular  to  the  principal  plane.  This 
is  a  pure  definition,  rather  than  a  statement  of  physical  fact, 
and  leaves  it  an  open  question  whether  the  vibrations  in  polar- 
ized light  lie  in  or  perpendicular  to  the  plane  of  polarization. 

123.  Elastic-solid  theories. — Such  a  question  could  not  be 
permanently  ignored  by  physicists,  particularly  as  it  was  found 
to  be  of  decisive  importance  for  certain    theories   which    were 
worked    out    mathematically    during    the    nineteenth    century. 
These  theories  all  started  from  the  assumption  that  the  ether 
behaves  like  an  elastic  jelly,  and  that    light    consists    of    real 
mechanical  transverse  waves  in  it,  the  velocity  of  which  de- 
pends on  the  density  and  the  elastic  coefficients  of  the  ether. 
Since  light  travels  slower  through  glass    and    other    material 
media  than  through  the  free  ether,  it  was  assumed  that  the 
presence  of  material  molecules  alters  either  the  density  or  the 
elastic  properties  of  the  ether.    In  doubly-refracting  substances, 
the  velocity  is  different  for  different  directions  of  the  vibra- 
tions, and  this  fact  would  lead  us  to  infer  that  it  is  the  elastic 
constants,  rather  than  the  density,  which  is  changed,  for  densi- 
ty, as  we  ordinarily  know  it,  is  not  a  vector  quantity,  and  has 

(240) 


ELECTROMAGNETIC  THEORY  241 

nothing  to  do  with  direction.  Nevertheless,  some  theorists  as- 
sumed that  it  is  the  density  that  is  changed,  others  that  it  is 
the  elastic  coefficients,  for  the  development  of  the  science  had 
reached  such  a  point  that  further  progress  could  be  made  only 
by  making  such  assumptions  and  seeing  whether  all  the  con- 
clusions to  which  they  led  were  in  accord  with  experimental 
facts.  Each  of  these  assumptions  accounts  for  some  of  the 
facts  of  reflection  and  double  refraction,  but  neither  is  satis- 
factory at  all  points.  One  leads  to  the  conclusion  that  in  polar- 
ized light  the  vibrations  are  parallel  to  the  plane  of  polariza- 
tion, the  other  that  they  are  perpendicular  to  it. 

A  fundamental  weakness  in  these  " elastic  solid"  theories 
is  that  they  fail  to  explain  why  we  never  find  indications  of 
longitudinal  ether  waves.  For  a  disturbance  inside  an  elastic 
solid  like  a  jelly  will  send  out  not  only  transverse  but  also 
longitudinal  waves,  which  will  travel  with  a  different  velocity, 
greater  or  less  than  the  transverse,  according  to  the  relative 
values  of  the  elastic  coefficients,  the  compressibility  and  the 
rigidity.  Moreover,  longitudinal  waves  striking  a  reflecting 
surface  should  set  up  transverse  waves,  and  vice  versa.  At- 
tempts were  made  to  solve  this  difficulty  by  assuming  that  the 
longitudinal  waves  failed  to  make  their  presence  known  to  us 
because  their  velocity  was  enormously  great  or  enormously 
small,  but  the  results  were  by  no  means  satisfactory.  It  is 
also  difficult  to  see  how  a  medium  which  exerts  an  elastic  re- 
action against  a  displacement  can  allow  bodies  like  the  planets 
to  move  through  it  without  any  retardation. 

124.  Electromagnetic  theory. — Although  text-books  writ- 
ten in  comparatively  recent  years  have  given  a  prominent  place 
to  the  elastic  solid  theories,  these  are  now  practically  obsolete, 
supplanted  by  the  electromagnetic  theory.  Therefore  we  shall 
in  this  book  make  no  further  reference  to  them,  but  confine  our 
attention  to  the  electromagnetic  theory  and  its  consequences.  A 
complete  mathematical  treatment  would  be  out  of  place  in  a 
text  of  this  kind,  but  an  attempt  is  made  in  the  following  pages 
to  give,  first  a  physical  conception  of  the  character  of  electro- 
magnetic waves,  second  a  statement  of  the  electromagnetic  laws 
that  form  their  basis,  with  a  little  of  the  history  of  the  develop- 
ment of  the  theory. 


242  LIGHT 

We  are  no  longer  to  think  of  the  ether  as  a  material  me- 
dium, with  a  certain  density  and  a  certain  degree  of  rigidity, 
but  simply  as  the  seat  of  electrical  and  magnetic  forces,  and 
the  laws  of  these  forces  are  all  that  we  need  to  know  about  it. 
Figure  123  is  designed  to  explain  the  electrical  part  of  what 


'\  t>\ 

- 


Figure    123 

we  mean  by  an  electromagnetic  wave.  At  ax,  a2,  etc.,  and  at 
points  in  their  immediate  neighborhood,  what  we  call  the  elec- 
tric force,  or  the  intensity  of  the  electric  field,  is  directed  up- 
ward in  the  figure,  while  at  bly  b2,  etc.,  and  in  their  neighbor- 
hood, it  is  directed  downward.  The  meaning  of  the  above  state- 
ment is  this  :  if  a  body  with  a  positive  electric  charge  were  placed 
at  ax  or  a2  it  would  be  acted  upon  by  an  upward  force,  while 
at  bj  or  b2  it  would  be  acted  upon  by  a  downward  force.  The 
small  vertical  arrows  indicate  the  direction  and  the  magnitude 
of  the  electric  force  at  the  different  points.  Now  imagine  this 
whole  condition  of  affairs  to  be  in  process  of  transference  to 
the  right  as  indicated  by  the  arrow  A,  so  that  after  a  certain 
time  the  a's  will  be  places  of  downward  and  the  b's  of  upward 
electric  force.  We  should  then  have  an  electric  wave  travelling 
in  the  direction  A. 

Such  a  wave,  however,  cannot  exist  alone,  for  the  changes 
in  electric  force  are  necessarily  accompanied  by  changes  in 
magnetic  force,  in  a  plane  at  right-angles  to  the  electric  force, 
which  means  at  right-angles  to  the  paper  in  figure  123.  To 
show  properly  both  the  electric  and  the  magnetic  parts  of  the 
complete  wave,  would  require  a  three-dimensional  model,  but  in 
figure  124  an  attempt  is  made  to  represent  the  whole  thing  in 


-'i      Ll    \--^- 


Figure    124 

a  perspective  drawing.  A  positive  charge  placed  at  at  or  a2 
would  experience  a  force  upward,  while  the  north-pointing  pole 
of  a  magnet  placed  there  would  experience  a  force  outward 


DIRECTION  OF  THE  VIBRATIONS  243 

toward  the  reader.  At  b,  or  b2  both  forces  would  be  reversed. 
The  reader  should  not  think  of  the  electric  and  the  magnetic 
parts  as  two  separate  waves,  but  simply  as  two  different  aspects 
of  the  same  wave,  for,  from  the  nature  of  electrical  and  mag- 
netic phenomena,  neither  one  can  exist  without  the  other. 

Such  a  wave  as  is  depicted  in  figure  124  would  undoubted- 
ly be  a  polarized  wave,  for  we  have  considered  the  electric 
vibrations  to  be  always  in  a  vertical,  the  magnetic  in  a  hori- 
zontal plane.  Either  set  of  vibrations  might,  however,  be  in 
any  plane  parallel  to  the  direction  of  propagation,  provided 
the  two  parts,  electric  and  magnetic,  are  perpendicular  to  one 
another.  A  train  of  waves  in  which  the  plane  of  the  electric 
vibrations  is  continually  changing  at  random  would  be  an  un- 
polarized  train.  Polarized  electromagnetic  waives  of  the  type 
of  figure  124  are  exactly  the  kind  sent  out  by  the  transmitting 
apparatus  of  a  wireless  telegraphy  outfit.  The  electric  vibra- 
tions are  perpendicular  to  the  earth's  surface,  the  magnetic 
parallel  to  it. 

The  electromagnetic  theory  of  light,  although  historically 
it  arose  long  before  the  development  of  wireless  telegraphy, 
really  amounts  to  the  belief  that  polarized  light-waves  are 
exactly  identical  in  everything  except  wavelength  and  fre- 
quency with  the  waves  of  wireless  telegraphy,  so  that  if  a  wire- 
less transmitter  could  be  made  to  vibrate  rapidly  enough,  and 
thus  give  out  short  enough  waves,  the  usual  receiving  appara- 
tus could  be  dispensed  with  and  the  signals  received  by  the  eye. 
Wireless  telegraphy  would  then  become  identical  with  signal- 
ing by  flashes  from  a  lantern,  and  some  of  its  advantages, 
whicK  depend  upon  great  length  of  wave,  would  be  lost. 

125.  Direction  of  the  vibrations. — The  introduction  of  the 
electromagnetic  theory,  with  its  two  kinds  of  vibrations  at 
right-angles  to  one  another,  makes  a  fundamental  change-  in 
the  old  question  about  the  plane  of  polarization.  Instead  of 
asking,  "Are  the  vibrations  parallel  or  perpendicular  to  the 
plane  of  polarization?",  we  must  now  ask,  "Is  it  the  electric 
or  the.  magnetic  vibrations  that  lie  in  the  plane  of  polariza- 
tion?" The  answer  to  this  question  has  been  found  by  apply- 
ing rigorous  mathematical  methods  to  the  reflection  of  elec- 
tromagnetic waves  from  glass  at  the  polarizing  angle,  first 
when  the  electric  vibrations,  second  when  the  magnetic,  lie  in 


244  LIGHT 

the  plane  of  incidence.  The  theory  indicates  that  in  the  -first 
case  there  is  no  reflection,  all  the  energy  being  transmitted, 
while  in  the  second  part  is  reflected  and  part  transmitted.  If 
the  incident  light  has  both  its  electric  and  its  magnetic  vibra- 
tions inclined  to  the  plane  of  incidence,  we  can  resolve  it  into 
two  components.  In  one  component,  the  electric  -vibrations  are 
in  the  plane  of  incidence  and  the  magnetic  perpendicular  to  it, 
and  no  part  of  this  component  is  reflected.  The  other  com- 
ponent has  its  magnetic  vibrations  in  the  plane  of  inci- 
dence and  its  electric  at  right-angles,  and  some  of  this 
energy  is  reflected.  Therefore,  in  general,  electromagnetic 
waves  reflected  at  the  polarizing  angle  have  their  mag- 
netic vibrations  in  the  plane  of  incidence,  and  their  elec- 
tric vibrations  at  right-angles.  Since  the  plane  of  polariza- 
tion has  been  defined  so  that,  for;  light  reflected  at  the 
polarizing  angle,  it  is  identical  with  the  plane  of  incidence,  we 
can  now  draw  the  general  conclusion  that,  if  the  electromag- 
netic theory  is  correct,  magnetic  vibrations  lie  in  the  plane  of 
polarization,  electric  at  right-angles  to  it.  Since  it  has  become 
customary  to  understand  when  one  speaks  of  the  ''vibrations" 
of  electromagnetic  waves,  without  any  qualifying  adjective, 
that  it  is  the  electric  vibrations  that  are  meant,  it  is  often 
stated  somewhat  loosely,  that  the  vibrations  are  perpendicular 
to  the  plane  of  polarization. 

The  theory  of  electromagnetic  waves  in  space  was  worked 
out  on  a  strictly  mathematical  basis  by  Clerk  Maxwell.  He 
was  able  to  prove  that  such  waves  are  possible,  and  that  their 
velocity  of  propagation  would  be  3  X  1010  centimeters  per 
second  in  the  free  ether.  Since  this  is  the  velocity  found  ex- 
perimentally for  light,  he  announced  the  belief  that  light  con- 
sists of  electromagnetic  waves  of  very  short  length. 

-A  complete  verification  of  Maxwell's  theory,  however, 
would  call  for  an  experimental  demonstration  that  waves  of  the 
character  contemplated  in  this  theory  actually  exist,  under 
such  circumstances  that,  from  the  manner  of  their  production 
and  the  methods  of  detecting  them,  there  could  be  no  doubt 
that  they  are  electromagnetic,  and  this  was  not  done  until 
after  Maxwell's  death.  The  difficulty  lay  not  simply  in  produc- 
ing the  waves, — for  according  to  Maxwell's  theory  any  elec- 
tric or  magnetic  phenomenon  such  as  the  discharge  of  a  con- 


FUNDAMENTAL  ELECTROMAGNETIC   LAWS     245 

denser  or  the  mere  movement  of  a  charged  body  or  magnet 
would  send  out  such  a  wave — but  in  getting  them  of  sufficient 
intensity  and  devising  suitable  means  of  detecting  them.  These 
experimental  difficulties  were  first  solved  by  Heinrich  Hertz, 
whose  work  formed  the  foundation  from  which  Marconi,  De- 
forest and  others  developed  modern  wireless  telegraphy. 

The  complete  mathematical  theory  of"  elec- 
tromagnetic waves  would  be  too  difficult  for 
a  book  such  as  this,  but  the  following  treat- 
ment should  give  some  physical  idea  of  why 
these  waves  are  possible  and  how  they  propa- 
gate themselves  through  the  ether.  We  begin 
by  recalling  two  fundamental  laws,  found  by 
experiment,  for  electromagnetic  phenomena. 

126.  Fundamental  electromagnetic  laws. — If  the  wire  AB 
in  figure  125  carries  an  electric  current  in  the  direction  shown 
by  the  arrow,  it  is  surrounded  by  a  "field  of  magnetic  force," 
that  is,  magnetic  lines  of  force  encircle  the  wire  in  the  direc- 
tion shown.     If  the  wire  is  long  and  straight,  and  the  return 
part  of  the  circuit  far  away,  the  lines  of  force  are  circles,  and 
in  fact  any  circle  coaxial  with  the  wire  is  a  line  of  force.   Other- 
wise, the  lines  are  not  true  circles,  but  they  still  form  closed 
curves  surrounding  the  wire.     The  direction  of  the  magnetic* 
lines  bears  the  same  relation  to  the  direction  of  the  current  as 
the  direction  of  rotation  of  a  right-handed  screw  to  the  direc- 
tion in  which  the  screw  advances.    This  is  our  first  fundamental 
empirical  law. 

The  second  law  relates  to  the  induction  of  electric  currents. 
Suppose  we  have  a  plain  loop  of  wire,  in  which  there  is  origi- 
nally no  current,  and  no  battery,  dynamo,  or  other  device  for 
producing  a  current.  Then  if,  by  any  such  means  as  the  move- 
ment of  a  magnet,  the  starting  or  stopping  of  a  current  in  the 
neighborhood,  etc.,  lines  of  magnetic  force  are  caused  to  thread 
the  loop,  or  lines  previously  threading  it  are  withdrawn,  a 
momentary  current  will  traverse  the  circuit. 

127.  Faraday's  displacement-currents. — Now  consider  the 
kind  of  circuit  which  we  call  incomplete,  shown  in  figure  126. 
The  two  straight  lines  at  C  represent  the  plates  of  a  condenser, 
such  as  the  inner  and  outer  coatings  of  a  Leyden  jar,  and  B  is 
an  ordinary  battery  cell.    A  steady  current  cannot  flow  in  such 


246 


LIGHT 


Figure  126 


a  circuit,  but  when  the  poles  of  the  cell  are  first  joined  by  con- 
necting wires  to  the  plates  of  the  condenser,  there  is  a  real 
current  which  lasts  only  a  very  short  time. 
The  current  is  clockwise,  that  is,  we  say  that 
positive  electricity  flows  from  the  right-hand 
plate  of  the  condenser  around  through  the 
wires  and  the  cell  to  the  left-hand  plate, 
leaving  the  right-hand  plate  negatively 
charged  and  charging  the  left-hand  plate 
positively.  The  current  ceases  when  the 
difference  in  potential  between  the  condenser 
plates,  due  to  their  charges,  equals  tha-t  be- 
tween the  poles  of  the  cell. 
Such  a  circuit  is  called  incomplete  because  there  is  no  con- 
ducting path  through  the  condenser.  But  Michael  Faraday 
took  the  view  that  every  electrical  circuit  is  in  a  certain  sense 
a  complete  circuit.  He  regarded  electricity  as  being  capable 
of  motion  within  a  non-conductor  (or,  dielectric)  as  well  as 
within  a  conductor,  but  with  this  difference :  in  a  conductor,  its 
motion  is  analogous  to  the  flow  of  water  through  a  pipe,  which 
retards  the  water  by  a  sort  of  frictional  resistance  but  has  no 
tendency  to  reverse  its  motion;  while  in  a  non-conductor  there 
is  a  sort  of  elastic  reaction  opposing  the  flow,  a  reaction  which 
becomes  greater  as  the  displacement  becomes  greater  and  which 
tends  to  reverse  it.  A  rather  good  analogy  with  Faraday's 
ideas  is  gotten  by  considering  the  wires  of  figure  126  replaced 
by  pipes,  the  cell  by  a  pump,  positive  electricity  by  water,  and 
the  condenser  by  an  elastic  membrane  which  the  water  cannot 
penetrate,  but  which  can  be  stretched  when  water  is  pumped 
away  from  one  side  of  it  to  the  other  side.  "When  the  pump 
is  started  water  flows  through  the  pipes  and  the  membrane  is 
bulged  out  toward  one  side,  and  the  flow  stops  when  the  mem- 
brane maintains  a  difference  of  pressure  equal  to  that  produced 
by  the  pump.  If  another  pipe-route,  not  containing  a  pump, 
is  opened  between  the  two  sides  of  the  membrane,  the  latter 
will  flatten  out,  discharging  the  water  through  this  route.  If 
the  resistance  of  this  pipe  is  not  too  great,  the  inertia  of  the 
water  will  cause  the  membrane  to  overshoot  the  mark,  it  will 
bulge  out  on  the  other  side,  and  there  will  be  a  series  of  oscil- 


THE  FARADAY-MAXWELL  THEORY  247 

lations  in  the  water,  till  it  is  brought  to  rest  through  loss  of 
energy  by  friction  in  the  pipes.  Analogous  electrical  oscilla- 
tions occur  when  a  condenser  is  suddenly  discharged  through 
a  wire  circuit  of  small  electrical  resistance.  Notice  that  the 
force  which  the  pipes  themselves  exert  upon  the  water  is  in 
the  nature  of  a  drag,  or  resistance.  It  increases  when  the 
velocity  of  flow  increases,  and  becomes  zero  when  the  velocity 
is  zero.  On  the  other  hand,  the  reaction  of  the  membrane  de- 
pends not  at  all  upon  the  velocity  of  flow,  but  only  upon  how 
much  water  has  been  pumped  out  from  one  side  into  the 
other. 

In  the  wires  of  the  electrical  circuit  there  is  a  real  current, 
or  flow  of  electricity,  when  the  condenser  is  being  charged  and 
when  it  is  being  discharged,  just  as  there  is  a  real  flow  of  water 
in  the  pipes  when  the  membrane  is  being  bulged  out  and* when 
it  is  flattening  itself  again.  Faraday  regarded  the  condenser 
also  as  the  seat  of  a  flow  of  electricity,  opposed  by  an  elastic 

reaction  instead  of  a  mere  resistance.     Such  a  flow  has  been 

• 

called  a  displacement  current,  as  distinguished  from  the  con- 
duction current  that  takes  place  in  the  metal  wires. 

128.  Maxwell's  assumptions. — So  far,  Faraday's  idea  is 
merely  a  theory  that  affords  a  convenient  way  of  thinking  of 
electrical  phenomena,  but  it  becomes  of  great  importance  when 
we  enquire  whether  displacement  currents  have  the  same  rela- 
tion to  magnetic  phenomena  as  conduction  currents.     Are  dis- 
placement currents  surrounded  by.  lines  of  magnetic  force? — 
Are  displacement  currents  induced  when  there  is  a  change  in 
the  lines  of  force  threading  through  a  circuit  composed  in  whole 
or  in  part  of  non-conductors?    An  answer  by  direct  experiment 
would  be  difficult,  but  Maxwell  started  out  with  the  assumption 
that  the  answer  to  each  of  these  questions  is  yes,  and  tried  to 
see  what  conclusions  this  assumption  would  lead  him  to.     The 
principal  conclusion  is  the  existence  of  electromagnetic  waves 
in  the  ether  of  the  kind  we  have  already  described. 

129.  Hertz's    experiments. — The    mechanism    by    which 
these  waves  propagate  themselves  can  be  seen  by  a  considera- 
tion of  the  apparatus  with  which  Hertz    first    studied    them. 
The  two  ends  of  the  secondary  of  an  induction  coil  (S,  figure 
127)  are  led  to  the  two  halves  of  what  is  called  the  oscillator. 
This  consists  of  two  rectangles  of  sheet  metal  in  the  same  plane, 


248 


LIGHT 


Figure    127 


to  each  of  which  is  attached  a  metal  rod  ending  in  a  ball.  The 
two  balls  are  placed  close  enough  together  so  that  when  the 
induction  coil  is  operated  a  series  of  sparks  pass  between  them. 

One  may  regard  the  rectan- 
\  gles  of  sheet  metal  as  being 

the  two  plates  of  a  condenser, 
\  the  plates  being  opened  and 
A  separated  from  one  another. 
;  They  still  constitute  a  sort  of 

condenser,   but  one  of  small 

capacity.  So  long  as  a  spark 
^''  is  in    existence    between    the 

balls,  the  spark,  the  balls  and 
.the  rods  form  a  short-circuit  for  the  condenser,  but  before  the 
spark  is  formed  the  plates  are  connected  only  through  the  coil 
S  in  the  induction  coil.  The  first  action  then,  when  the  induction 
coil  starts  to  work,  will  be  to  charge  up  the  plates  just  as  if  the 
induction  coil  were  a  battery  cell,  except:  that  they  are  charged 
to  much  higher  potential.  Let  us  say,  for  the  sake  of  concrete- 
ness,  that  the  upper  plate  is  charged  positively,  the  lower  nega- 
tively. Evidently  there  will  be  what  we  have  called  displacement- 
currents  in  the  space  between  the  balls  and  also  all  around  the 
two  plates.  When  the  difference  in  potential  between  the  two 
plates  becomes  great  enough,  a  spark  starts  between  the  balls, 
immediately  establishing  a  short-circuit.  The  plates  then  dis- 
charge, and  rapid  electrical  oscillations  occur  in  the  system, 
much  more  rapid,  in  fact,  than  the  operation  of  the  induction 
coil  which  starts  them.  The  induction  coil,  for  example,  might 
make  one  or  two  hundred  sparks  per  second,  while  the  oscilla- 
tions in  the  oscillator  are  at  the  rate  of  some  hundred  million 
per  second,  so  that  each  single  spark  would  include  a  large 
number  of  oscillations,  though  in  each  set  of  oscillations  there 
is  no  doubt  that  the  amplitude  dies  down  rapidly,  or,  as  we 
say,  they  are  strongly  damped.  Each  spark  starts  a  series  of 
oscillations  which  die  out  completely  before  the  next  spark 
occurs. 

130.  Propagation  of  electromagnetic  waves. — Now,  con- 
sider what  happens  in  the  surrounding  space,  supposing  for 
convenience  that  the  axis  of  the  oscillator  is  vertical.  When 
the  first  discharge  occurs,  there  is  a  downward  current  through 


PROPAGATION  OF  THE  WAVES  249 

the  spark  which  causes  lines  of  magnetic  force,  clockwise  as 
seen  from  above,  to  circulate  in  a  horizontal  plane  about  the 
oscillator.  According  to  our  second  law  of  electromagnetic 
phenomena,  the  introduction  of  magnetic  lines  through  any 
conducting  circuit  will  induce  a  momentary  current  in  the 
latter;  and  according  to  Maxwell's  theory  the  same  thing  holds 
true  also  for  a  non-conducting  circuit.  Therefore,  any  circuit 
drawn  at  will  so  as  to  enclose  some  of  these  lines  of  magnetic 
force,  such  as  the  dotted  line  in  the  figure,  which  is  drawn  to 
represent  a  circle  in  a  vertical  plane,  would  be  the  seat  of  an 
induced  current,  whose  direction  is  given  by  the  arrow.  This 
means  that  there  would  be  an  induced  current  upward  at  the 
spark  (where  its  effect  would  be  to  diminish  somewhat  the 
downward  current  already  there,  and  so  account  in  part  for 
the  dying  down  of  the  amplitude  of  the  oscillations)  and  down- 
ward at  A.  This  downward  current  at  A  would  in  its  turn 
produce  lines  of  magnetic  force  in  a  horizontal  plane,  clock- 
wise as  seen  from  above.  Their  effect  would  be  to  neutralize 
the  magnetic  field  between  A  and  the  oscillator,  and  to  produce 
beyond  A  other  induced  displacement  currents,  which  would  in 
turn  produce  a  magnetic  field,  and  so  on.  Thus,  the  effect  of 
the  downward  current  in  the  oscillator  is  to  cause  downward 
displacement  currents  and  lines  of  magnetic  force  in  a  horizon- 
tal plane,  which  progress  farther  and  farther  from  the  oscil- 
lator. The  above  explanation  might  lead  the  reader  to  draw 
the  incorrect  conclusion  that  the  progression  out  from  the  oscil- 
lator comes  in  a  series  of  steps,  for  an  explanation  of  such  a 
complicated  phenomenon  given  without  the  assistance  of  mathe- 
matical analysis  is  necessarily  crude  and  incomplete.  In  reality 
the  progression  is  perfectly  continuous  and  goes  on  at  a  veloc- 
ity of  which  we  shall  speak  later. 

The  downward  discharge  of  the  condenser  is  followed  after 
a  very  small  fraction  of  a  second  by  a  reverse  discharge  up- 
ward. This  of  course  will  also  send  out  vertical  displacement- 
currents  and  horizontal  magnetic  field,  both  of  which  will  be 
in  the  reverse  direction  from  the  corresponding  vectors  caused 
by  the  initial  discharge,  but  they  will  follow  them  in  their 
progression  out  from  the  oscillator.  The  phenomena  are  re- 
peated by  the  succeeding  discharges,  although  the  amplitude  of 
the  oscillations  must  become  less  and  less  since  energy  is  con- 


250  LIGHT 

stantly  being  sent  out  from  the  oscillator.  Thus  there  will  be 
a  series  of  damped  electromagnetic  waves,  for  each  time  that 
the  induction  coil  charges  up  the  plates  of  the  oscillator.  It 
is  evident  that  the  waves  are  transverse  and  polarized,  since 
electrical  vibrations  occur  only  in  a  vertical,  magnetic  only  in 
a  horizontal  plane. 

We  have  traced  the  causes  of  the  wave  propagation  only 
for  a  direction  at  right  angles  to  the  axis  of  the  oscillator, 
but  waves  are  also  sent  out  in  other  directions,  though  with 
less  intensity.  No  waves  at  all  are  sent  out  in  a  direction 
parallel  to  the  axis. 

131.  Velocity  of  the  waves. — The  general  method  which 
we  have  here  used  to  show  how  electromagnetic  waves  are  prop- 
agated does  not  enable  us  to  predict  the  velocity  of  propaga- 
tion. Evidently  this  will  depend  upon  the  electrical  and  mag- 
netic properties  of  the  free  ether,  or  of  whatever  medium  the 
waves  are  passing  through.  The  mathematical  treatment  shows 
that  the  velocity  is 


where  /*  is  the  magnetic  permeability  of  the  medium  and  k  is 
its  dielectric  constant  (specific  inductive  capacity).  In  order- 
to  calculate  the  velocity  from  this  formula,  it  is  of  course  nec- 
essary that  both  p  and  k  be  expressed  in  the  same  system  of 
units,  for  two  systems  are  used  in  the  theory  of  electromagnetic 
phenomena,  the  electrostatic  system  and  the  electromagnetic. 
Since  we  must  choose  one  or  the  other,  suppose  the  latter  is 
chosen,  so  that  p,  and  k  are  both  expressed  in  electromagnetic 
units.  Now  the  electromagnetic  system  is  based  on  the  assump- 
tion that  the  permeability  of  the  ether  is  1,  the  electrostatic 
system  on  the  assumption  that  the  dielectric  constant  of  the 
ether  is  1 ;  therefore  in  the  above  equation  for  the  velocity  we 
may  substitute  1  for  ^  but  we  may  not  substitute  1  for  k. 
Instead,  we  must  find  the  value  of  k  in  electromagnetic  units. 
When  two  charges  e  and  e'  are  placed  x  centimeters  apart,  the 
force  between  them  is 

v  - ee' 

~.Sf 

Since  k  is  expressed  in  electromagnetic  units,  e  and  e'  must  be 


VELOCITY  ^OF  PROPAGATION  251 

also.    If  E  and  E'  be  the  values    of    these    same    charges    in 
electrostatic  units, 


where  K  is  now  the  dielectric  constant  in  the  electrostatic  sys- 

tem.   F  and  x  remain  as  they  were,  for  the  unit  of  force  and 

the  unit  of  distance  do  not  change  from    one    system    to  the 
other.     Therefore,  we  may  say 

ee'__  EE' 
k  r      iv 

Now  let  c  be  the  number  of  electrostatic  units  of  charge  which 
are  equal  to  one  electromagnetic  unit  of  charge.     Then 
E  =  ce  E'  =  ce' 

ee'      c2ee' 


k=| 

Since  K  =  1  for  the  ether,  we  may  write  k  =  1/c2.  Substi- 
tuting this  value  for  k  and  1  for  ^  in  the  equation  for  the 
velocity,  we  get 

V  =  c 

The  velocity  of  electromagnetic  waves  appears  then  to  be  equal 
to  the  ratio  of  the  units  of  charge  on  the  two  systems  of  elec- 
trical units.  This  ratio  has  been  measured  with  great  care  by 
a  number  of  different  experimental  methods,  and  is  found  to 
be  3  X  1010,  the  same  value  as  the  velocity  of  light  in  centi- 
meters per  second.  This  is  also  the  value  found  experimentally 
by  Hertz  for  the  velocity  of  his  electromagnetic  waves. 

From  this  close  accord  between  Hertz's  experimental  re- 
sults and  Maxwell's  theory  on  the  one  hand,  and  the  known 
properties  of  light  on  the  other,  the  electromagnetic  theory  of 
light  seems  fairly  surely  grounded.  One  is  naturally  led  to 
enquire  what  sort  of  a  mechanism  sends  out  the  extremely 


252  LIGHT 

short  waves  that  constitute  visible  light.  Light  always  starts 
from  some  material  particles,  never,  we  believe,  from  an  empty 
spot  in  the  ether.  Moreover,  we  have  seen  that  the  different 
chemical  elements  emit  characteristic  wavelengths.  Consequent- 
ly, the  mechanism  of  emission  for  ordinary  light,  corresponding 
to  the  oscillator  of  Hertz  or  the  elaborate  transmission  appara- 
tus of  wireless  telegraphy,  must  be  contained  within  the  atom. 
Many  phenomena  not  directly  connected  with  light  indicate 
that  an  atom  contains  positive  and  negative  charges,  and  we  are 
led  to  infer  that  it  is  the  vibrations  of  one  or  more  of  these 
charges  which  start  the  electromagnetic  waves.  A  mass  of 
material  becomes  luminous  whenever,  as  a  result  of  high  tem- 
perature or  any  other  cause,  the  charges  within  some  of  the 
atoms  are  set  into  vibration  with  sufficient  amplitude  to  pro- 
duce, in  the  surrounding  ether,  waves  strong  enough,  to  affect 
the  eye.  Waves  coming  from  a  single  atom  would  be  polarized, 
but  from  a  great  mass  of  luminous  atoms,  as  in  a  flame,  all 
planes  of  vibrations  would  be  represented.  If  the  waves  strike 
against  any  material  body,  part  of  thei?  energy  is  reflected  and 
part  enters.  In  some  materials  the  waves  that  enter  are  ab- 
sorbed before  they  have  gone  more  than  a  short  distance,  their 
energy  being  converted  into  heat.  Such  materials  are  said  to 
be  opaque  to  the  waves.  Other  materials,  called  transparent, 
allow  the  waves  to  pass  through  with  little  absorption.  The 
velocity  is  always,  with  a  few  exceptional  cases,  slower  in  the 
material  than  in  the  free  ether;  and  if  the  material  is  doubly- 
refracting  there  are  two  velocities — one  for  waves  having  the 
electrical  vibrations  in  the  principal  place  (extraordinary), 
and  one  for  those  having  the  electrical  vibrations  perpendicular 
to  the  principal  plane  (ordinary). 

132.  Refractive  index  and  dielectric  constant. — Of  course, 
since  V  =  l/\/ju,k,  a  change  in  velocity  means  a  change  in  p 
or  in  k.  The  permeability  for  all  materials  except  the  mag- 
netic metals,  iron,  nickel,  cobalt,  etc.,  is  practically  the  same 
as  for  the  free  ether,  so  that  it  must  be  a  change  in  k,  the 
dielectric  constant,  which  causes  the  change  in  velocity.  In 
fact,  the  index  of  refraction  should  bear  a  direct  relation  to 
the  dielectric  constant.  For  if  V  is  the  velocity  of  the  light 
in  the  material  in  question,  V  that  in  the  ether,  k  the  dielec- 


REFRACTIVE  INDEX  253 

trie  constant  for  the  material  and  k'  that  for  the  ether,  the 
index  of  refraction  will  be 

X_'_  -L      L     \/F 

~~      -        '  '  V  A  =    ?  P 


Since  we  have  to  do  only  with  the  ratio  of  two  dielectric  con- 
stants, it  is  immaterial  whether  we  use  the  electromagnetic  or 
the  electrostatic  units.     If  we  use  the  electrostatic,  k'   =   1, 
n.  =  \/k,  or  na  =  k.     That  is,  the  square  of  the  refractive 
index  is  equal  to  the  dielectric  constant  in  electrostatic  units. 
The  following  table  shows  how  well  this  prediction  of  theory 
agrees  with  experimentally  measured  values  for  a  few  common 
materials.    The  refractive  indices  are  given  for  yellow  light  of 
wavelength  .00005893  cm. 

n        n2        k 
Carbon  bisulphide    ............    1.64    2.68     2.62 

Turpentine  oil   ................  1.47     2.16     2.23 

Benzene    .....................    1.50     2.25     2.29 

Ice   .........................    1.31     1.71     2.85 

Alcohol   ......................    1.37     1.88  25.8 

Water    ....................  ...    1.33     1.78  81 

It  will  be  noted  that  n2  and  k  are  nearly  the  same  in  value  for 
carbon  bisulphide,  turpentine  oil,  and  benzene,  but  quite  differ- 
ent for  ice  and  absurdly  different  for  alcohol  and  water.  This 
failure  of  agreement  causes  a  suspicion  that  there  is  some  de- 
fect in  the  theory,  a  suspicion  that  becomes  all  the  greater  when 
one  reflects  that  even  for  a  single  substance  the  index  varies 
with  wavelength,  while  the  dielectric  constant,  as  we  have  de-  / 
fined  it,  has  nothing  to  do  with  wavelength.  The  defect  ac-  ' 
tually  lies,  not  in  the  electromagnetic  theory  of  light,  but  in 
the  conception  of  the  dielectric  constant  for  material  substances. 
This  quantity,  which  is  defined  for  steady  electrostatic  fields, 
requires  some  modifications  when  applied  to  such  rapidly  alter- 
nating fields  as  we  are  concerned  with  in  light  waves.  These 
modifications  will  be  considered  in  the  next  chapter. 


CHAPTER  XV. 

133.  Dispersion.—  134.  Electron  theory  of  matter.—  135.  Electro- 
magnetic dispersion  formula.  —  136.  Anomalous  dispersion.  —  137.  Rest- 
strahlen. 

133.  Dispersion.  —  The  preceding  chapter  ended  with  the 
citation  of  a  case  where  the  electromagnetic  theory,  in  its  sim- 
ple form,  disagrees  decidedly  with  experiment.  According  to 
the  theory,  the  index  of  refraction  of  water  should  be  equal 
to  the  square-root  of  its  dielectric  constant,  i.  e.,  about  9,  where- 
as it  is  in  fact  about  1.33  for  yellow  light  and  variable  for 
different  wavelengths.  Similar  wide  divergences  are  shown  by 
some  other  materials,  and  the  index  varies  with  the  wavelength 
in  all.  Obviously,  before  the  electromagnetic  theory  can  be 
made  worthy  of  acceptance,  it  must  be  supplemented  in  some 
way  to  account  for  dispersion,  or  the  variation  of  index  with  v 
wavelength.  So  far  as  the  passage  of  light  through  the  ether 
is  concerned,  the  theory  may  stand  as  it  is,  for  it  is  only  in 
ponderable  matter,  made  up  of  molecules  and  atoms,  that  the 
velocity  is  different  for  different  colors,  or  wavelengths. 

No  two  materials  behave  exactly  alike  in  regard  to  dis- 
persion, but  so  far  as  the  visible  spectrum  is  concerned  most 
of  them  agree  in  this,  that  both  the  index  and  the  rate  at  which  / 
the  index  changes  are  greater  for  short  than  for  long  waves. 
Cauchy  proposed  the  formula 


where  A,  B,  and  C  are  constants  for  one  material,  but  have 
different  values  for  different  materials.  The  formula  is 
rather  useful  when  applied  to  prisms  of  glass,  etc.,  for 
if  we  know  the  indices  of  a  particular  glass  for  three 
different  wavelengths,  then  by  substituting  the  values  of 
A  and  the  corresponding  values  of  11  we  get  three  separate 
equations  in  which  A,  B,  and  C  occur  as  unknowns.  If  we 
solve,  and  substitute  the  numerical  values  in  (1),  we  have  an 
equation  from  which  n  can  be  found  for  other  wavelengths. 
But  the  equation  works  satisfactorily  only  over  a  limited  range 
of  wavelengths,  and  has  no  sound  theoretical  basis. 

(254) 


ELECTRON  THEORY  255 

134.  Electron  theory  of  matter. — The  dispersion  of  ma- 
terials can  be  accounted  for  on  the  electromagnetic  theory,  and 
the  nature  of  the  dielectric  constant  at  the  same  time  explained, 
if  we  take  into  account  the  electron  theory  of  matter,  brought 
into  being  as  a  result  of  researches  partly  in  optics,  partly  in 
radioactivity  and  other  branches  of  physics.     According  to  it, 
every  atom  consists  of  a  nucleus  of    positive    electricity    sur- 
rounded by  a  number  of  very  small  negative  charges  called 
electrons.    All  the  negative  electrons  are  believed  to  be  exactly 
alike,  even  in  different  elements,  and  the  elements  differ  merely 
in  the  number  and  arrangement  of  electrons  and  the  correspond- 
ing magnitude  of  the  positive  nucleus.     In  a  complete  atom, 
the  sum  of  all  the  negative  charges  equals  the  positive  charge 
of  the  nucleus,  so  that  the  atom   is    electrically    neutral,    but 
under  some  circumstances  one  or  more  electrons  may  become 
detached  from  the  atom  and  either  remain  free  and  unattached 
in  space  or  become  temporarily  attached  to  some  other  atom 
or  group  of  atoms.    Such  a  free  electron,  or  one  attached  to  an 
otherwise  neutral  atom  or  group  of  atoms,  is  called  a  negative 
ion;  while  the  atom  fromi  which  it  came,  which  is  left  with 
an  excess  of  positive  electricity,  and  which  possibly  also  gathers 
neutral  atoms  about  it,  is  a  positive  ion.     Experiments  on  the 
leakage  of  electricity  through    gases    indicate    that    there    are 
always  a  number  of  ions  present  in  any  gas,  though  exceedingly 
few  in  comparison  with  the  number  of  neutral  atoms,  and  that 
certain  agencies  such    as    X-rays    and    radioactive    substances 
cause  a  considerable  increase  in  the  number  of  ions.    In  metals 
the  electrons  are  believed  to  be  particularly  free  to  become  de- 
tached from  their  atoms  and  susceptible  of  great  freedom  of 
motion  within  the  metal.    This  accounts  for  the  fact  that  metals 
are  invariably  good  conductors  of  electricity,  for  the  electrons 
are  supposed  to  carry  the  current.     In  nonconductors,  on  the 
other  hand,  the  electrons  are  not  easily  detached  from  the  atoms. 

135.  Electromagnetic   dispersion  formula. — The  dielectric 
constant,  or  specific  inductive  capacity,  is  defined  as  follows: 
Imagine  two  small  metal  balls,  one  charged  positively,  the  other 
negatively,  e  and  e'    being  the  respective  charges  and  r  the 
distance  apart.     Since  the  force  of  attraction    is    proportional 
to  the  product  of  the  charges  and  inversely  as  the  square  of 
the  distance,  we  can  write  it 


256  LIGHT 


"~kr» 

The  factor  of  proportionality,  k,  is  the  dielectric  constant,  and 
its  value  depends  upon  the  nature  of  the  medium  between  and 
immediately  surrounding  the  charges.  If  the  medium  is  water, 
F  has  quite  a  different  value  from  that  for  the  ether,  and 
therefore  k  is  different.  According  to  the  electron  theory, 
this  decided  alteration  in  the  force,  caused  by  interposing 
material  atoms,  is  due  to  the  production  of  what  is  called  elec- 
trical polarization.  In  each  atom  the  positive  nucleus  is  pulled 
slightly  toward  the  negatively  charged  ball,  the  negative  elec- 
trons toward  the  positive  ball.  When  the  atoms  are  of  such  a 
nature  that  this  polarization  is  great,  the  result  will  be  a  value 
of  k  differing  greatly  from  that  for  the  ether. 

If  the  amount  of  polarization  is  proportional  to  the  in- 
tensity of  the  electrical  field  producing  it,  k  should  be  constant, 
and  we  find  that  it  is  so  for  steady  electrostatic  fields.  But 
when  light  waves  pass  through  a  material  the  latter  is,  as  we 
have  seen,  the  seat  of  electrical  fields  which  alternate  very 
rapidly,  and  in  such  a  case  we  should  expect  the  so-called 
dielectric  constant  not  to  be  constant  at  all,  but  highly  varia- 
ble. Particularly  should  this  be  the  case  if  the  waves  happened 
to  have  a  period  somewhere  near  a  natural  period  for  the 
vibration  of  the  electrons  within  the  atoms ;  that  is,  if  the  wave- 
length happened  to  be  nearly  the  same  as  that  which  the  atoms 
could  absorb  very  strongly,  or  could  themselves  emit  if  excited 
to  luminescence.  In  such  a  case  the  wavelength  is  said  to  lie 
close  to  an  absorption-band  of  the  material.  The  electrons 
would  be  set  into  violent  vibration  by  resonance,  and  the  elec- 
trical polarization  would  oscillate  through  a  very  wide  range. 

In  the  theory  of  such  a  case,  it  is  necessary  to  calculate 
a  sort  of  average,  or  effective,  value  for  the  dielectric  constant, 
which  will  of  course  be  different  for*  different  wavelengths. 
The  calculation  is  complicated,  because  we  must  take  into  ac- 
count not  only  the  amplitude  of  the  electronic  vibrations,  but 
also  the  relation  in  phase  between  these  vibrations  and  the  light 
vibrations.  For  instance,  if  the  period  of  the  light-waves  is 
shorter  than  the  natural  period  of  the  electronic  vibrations, 
the  latter  will  be  behind  the  former  in  phase,  and  conversely 


DISPERSION  FORMULA 


257 


in  the  converse  case.  The  full  mathematical  treatment  is  be- 
yond the  scope  of  this  text,  but  the  above  considerations;  are 
sufficient  to  show  that  the  effective  value  of  k,  and  therefore 
the  value  of  n,  should  be  abnormally  large  or  small  for  wave- 
lengths close  to  an  absorption-band.  When  the  atoms  have 
only  one  such  band,  the  final  formula  is 


M,  (A2  — 


(2) 


II2  =  1  + 


'    (A*_x,*)*  + 

Here  n  is  the  index  for  light  of  wavelength  A,  Aj  is  the  wave- 
length which  the  material  absorbs  most  strongly,  and  b  and  Mx 
are  constants  whose  values  depend  upon  conditions  within  the 
atom  and  the  density  of  the  material.  If  the  atom  has  several 
absorption  bands,  the  formula  becomes 

M2  (A2  — A,-) 


MT  (A2  — A,2) 


M.,  (A2-A.,2) 


(A2  —  ^T  +  V  '  A2  — A2  2  (A»^^)44-V 

Aj,  Ao.  A3,  ete.,  being  the  wavelengths  of  the  several  absorption 
bands. 

136.  Anomalous  dispersion. — If,  taking  equation  (2),  we 
plot  n  as  ordinate  against  A  as  abscissa,  we  get  a  curve  like 
figure  128,  indicating  very  low  values  of  n  for  waves  slightly 


Figure    128 

shorter  than  A17  very  large  values  for  waves  slightly  longer. 
There  are  certain  substances,  particularly  the  aniline  dyes, 
which  have  very  strong  absorption  bands  within  the  visible 
region,  and  for  these  the  dispersion  curves  are  found  by  experi- 
ment to  be  of  the  form  of  this  figure,  except  that  the  exceed- 


258  LIGHT 

ingly  strong  absorption  which  these  substances  exert  upon  light 
whose  wavelength  comes  anywhere  close  to  the  center  of  the 
absorption  band  (i.  e.,  close  to  A±  in  the  formula)  prevents  us 
from  taking  measurements  there.  For  instance,  all  that  part 
of  the  graph  which  is  indicated  in  dotted  lines  might  be  miss- 
ing from  the  experimental  curve.  Such  substances  were  said  to 
have  ''anomalous  dispersion,"  because  the  curve,  with  two 
branches,  differs  so  much  from  the  single  curve  obtained  for 
colorless  transparent  materials  in  the  visible  region.  We  now 
know,  however,  that  the  complete  dispersion  curve  of  any 
material,  taken  over  the  complete  range  of  wavelengths,  would 
show  an  inflection  like  that  of  figure  128  at  each  strong  absorp- 
tion band.  The  curve  for  the  visible  region,  in  the  case  of 
different  kinds  of  glass,  quartz,  etc.,  is  similar  to  the  right- 
hand  branch  of  the  figure,  showing  that  these  materials  must 
have  absorption-bands  in  the  ultraviolet. 

There  are  some  cases  of  absorption  which  do  not  seem  to 
influence  dispersion  in  this  way,  for  example  the  absorption 
of  solutions  of  copper  sulphate,  potassium  bichromate,  etc. 
Such  absorption  is  much  weaker,  however,"  than  the  kind  shown 
by  the  aniline  dyes. 

A  very  interesting  case  of  absorption  as  influencing  dis- 
persion is  that  of  sodium  vapor,  which  absorbs  very  strongly 
two  wavelengths  quite  close  together  in  the  yellow.  (See  sec- 
tions 53  and  56).  It  is  possible  to  adjust  a  Bunsen  flame  fed 
with  sodium  so  that  it  takes  the  form  of  a  prism,  and  thus 
investigate  its  index  of  refraction  for  light  of  different  wave- 
lengths. Since  the  density  of  the  vapor  is  quite  small,  the  in- 
dex is  practically  unity  except  for  wavelengths  very  close  to 
one  or  the  other  of  these  two  absorption  lines,  that  is  light 
passes  through  it  with  practically  the  same  speed  as  through 
air.  An  ingenious  experiment  due  to  Becquerel  causes  the 
vapor  to  draw  its  own  dispersion  curve.  Suppose  the  sodium 
flame  to  be  made  to  take  a  prismatic  form,  with  the  refracting 
edge  of  the  prism  below  and  horizontal,  and  placed  in  front 
of  the  slit  of  a  grating  spectroscope.  Light  from  some  source 
giving  a  continuous  spectrum  is  sent  through  this  flame,  enters 
the  slit  of  the  spectroscope,  and  is  spread  out  by  the  grating 
into  a  spectrum  in  a  horizontal  plane.  Those  wavelengths  for 
which  the  sodium  vapor  has  a  high  index  of  refraction  will  be 


RESTSTRAHLEN  259 

bent  upward  by  the  flame-prism,  and  so  will  be  raised  above  the 
general  level  of  the  spectrum,  while  those  for  which  the  index 

is  low  will  be  sent  down- 
absorption 

ward  and  brought  below  lincs 

the  general  level.  Figure 

129  shows  the  actual  ap-     Vlole*  — 


pearance  under  such  cir- 
cumstances.    Each  of  the  Figure  129 
absorption  lines  produces  a  break  in  the  dispersion-curve  simi- 
lar to  that  shown  in  figure  128,  affording  a  very  satisfactory 
verification  of  the  theory. 

137.  Reststrahlen.— There  are  a  large  number  of  cases 
where  a  material  has  the  power  to  resonate  to  a  certain  period 
of  vibration  so  that  we  should  expect  it  to  absorb  very  strongly 
light  of  the  corresponding  wavelength,  but  instead  it  reflects 
this  wavelength  with  extraordinary  power,  in  some  instances 
as  strongly  as  a  silvered  mirror  reflects  ordinary  visible  light. 
The  reflection  in  such  cases  is  a  true  surface  reflection,  not  the 
sort  of  diffuse  reflection  from  within  the  material  that  occurs 
in  the  case  of  paper  and  many  other  more  or  less  translucent 
materials,  and  which,  coupled  with  some  absorption,  is  responsi- 
ble for  the  colors  of  most  natural  objects.  This  selective  sur- 
face reflection,  due  to  resonance,  is  particularly  common  among 
substances  having  strong  resonance  bands  far  out  in  the  infra- 
red, such  as  quartz,  rocksalt,  sylvite,  potassium  bromide,  potas- 
sium iodide,  and  other  crystals  or  fused  salts.  This  property 
enables  us  to  isolate  and  study  some  very  long  waves  which 
would  otherwise  be  difficult  to  detect.  The  method  employed 
is  to  take  the  radiation  from  some  source  such  as  a  Welsbach 
mantle  burner  and  reflect  it  again  and  again  from  polished 
surfaces  of  the  material  being  investigated.  The  light  obtained 
from  a  single  reflection  contains,  beside  the  wavelength  selec- 
tively reflected,  light  of  other  wavelengths  which  are  reflected  to 
a  lesser  degree,  but  the  latter  are  almost  completely  eliminated 
after  a  number  of  reflections,  while  the  former  remain  almost  as 
strong  as  at  first.  The  Germans  have  called  waves  isolated  in 
this  manner  "Reststrahlen,"  and  this  name  has  been  pretty  gen- 
erally adopted  into  other  languages,  though  the  English  equiva- 
lent "residual  rays"  is  also  used.  By  this  means  light-waves  as 


260  LIGHT 

long  as  .00965  cm.  (nearly  1/10  mm.,  more  than  100  times  as 
long  as  the  deep  red)  have  been  isolated  from  the  radiation 
of  a  Welsbach  burner  by  using  potassium  iodide  for  the  re- 
flecting material. 

The  student  may  wonder  why,  if  these  wavelengths  are 
already  present  in  the  light  from  a  Welsbach  burner,  they 
could  not  be  detected  easily  and  simply  by  sending  the  whole 
beam  through  a  prism,  or  dispersing  it  with  a  grating,  as  we 
would  for  shorter  wavelengths,  without  resorting  to  the  resid- 
ual ray  method  for  first  isolating  them.  Prisms  are  useless 
for  such  a  purpose  because  many  prisms  would  absorb  such 
long  waves,  and  even  if  that  did  not  occur  we  could  hardly 
hope  to  determine  wavelengths  from  the  deviation  by  the  prism 
because  the  course  of  the  prism's  dispersion  curve  so  far  in 
the  infrared  would  not  be  accurately  known.  The  trouble  with 
using  a  grating  lies  in  the  fact  that  a  grating  gives  many 
spectra  which  overlap,  and  a  wavelength  such  as  we  are  con- 
cerned with  would  come  in  the  same  place  as  many  other 
shorter  waves.  For  example,  a  wavelength  of  .00965  in  the 
first  spectrum  would  coincide  with  .004825  in  the  second,. 
.003217  in  the  third,  .002412  in  the  fourth,  and  many  others, 
and  it  would  be  impossible  from  such  a  complex  to  tell  what 
wavelengths  were  actually  present.  After  the  method  of  re- 
flections from  a  selectively  reflecting  material  have  been  used 
to  isolate  the  waves,  a  coarse  grating  or  a  specially  constructed 
interferometer  can  be  used  to  measure  the  wavelengths.  Of 
course  the  actual  detecting  device  must  be  a  bolometer,  ther- 
mopile, or  some  similar  absorbing  and  heat-recording  instru- 
ment. 

It  was  found  that  quartz,  although  opaque  to  some  shorter 
waves,  is  rather  transparent  to  the  residual  rays  from  potas- 
sium iodide,  and  has  for  such  waves 
the  very  large  refractive  index  2.2. 
A  quartz  lens  then  would  have  for 
such  long  waves  a  focal  length  much 
less  than  for  shorter  waves.    Rubens 
Figure  130  and  Wood  made  use  of  this  fact  for 

isolating  these  waves  and  even  longer  ones,  their  method 
being  in  principle  as  shown  in  figure  130.  A  is  a  Wels- 


LONGEST  INFRARED  WAVES.  261 

bach  mantle,  emitting  the  radiations,  B  a  double  metal 
screen  with  a  small  hole,  C  a  quartz  lens,  E  another  screen 
with  a  small  hole.  E  is  placed  at  the  proper  focus  for 
the  very  long  waves  to  be  investigated.  For  visible  and 
the  shorter  infrared  waves,  the  focal  length  of  the  lens  is 
so  long  that  the  hole  in  B  comes  within  the  principal  focus. 
Therefore  such  waves  are  diverged  by  the  lens,  and  none  of 
them  would  get  through  the  hole  in  E  except  such  as  came 
through  close  to  the  center  of  the  lens,  and  these  were  cut  out 
by  fastening  there  a  small  opaque  disc  D.  It  is  clear  that  by 
this  method  not  only  the  particular  waves  strongly  reflected  by 
potassium  iodide  would  get  through,  but  also  other  waves  hav- 
ing about  the  same  index  of  refraction,  which  would  be  lost  in 
the  reststrahlen  method.  In  this  way,  they  were  able  to  obtain 
waves  as  long  as  .0107  cm.  from  a  Welsbach  mantle,  and  by 
using  a  mercury  arc  in  a  quartz  tube  instead  of  the  Welsbach 
burner  Rubens  and  von  Baeyer  detected  waves  of  .0343  cm., 
the  longest  waves  yet  found  in  the  radiation  from  a  light- 
source.  Waves  from  electrical  oscillations  in  constructed  ap- 
paratus (i.  e.,  waves  of  the  Hertz  type)  have  been  obtained 
as  short  as  .2  cm.,  leaving  only  a  very  short  gap  for  undetected 
wavelengths.  Doubtless  waves  of  length  suitable  to  fill  in  this 
gap  exist,  and  it  only  remains  to  detect  and  measure  them. 

Problems. 

1.  Calculate  the  constants  A,  B,  and  C,  of  equation   (1) 
of  the  preceding  chapter,  for  glass  whose  indices  are  1.7774  for 
A  .00004713,  1.7582  for  A  .00005600,  and  1.7444  for  A  .00006708. 
Then    calculate    the    indices    for    wavelengths    .00005016    and 
.00005893. 

2.  Show  that  the  middle  point  in  a  spectrum  as  given  by 
a  prism  of  the  glass  of  problem  1  would  correspond  to  a  much 
shorter  wavelength  than  that  in  a  spectrum  of  the  same  source 
as  given  by  a  grating. 

3.  From  a  consideration  of  the  above  two  problems  show 
why,  in  general,  spectra  produced   by    prisms    are,    relatively 
speaking,  abnormally  bright  in  the  longer  wavelengths. 


CHAPTER  XVI. 

138.  Production  of  X-rays.— 139.  Their  properties.— 140.  Are  X-rays 
ether  waves? — 141.  Crystal  reflection  of  X-rays. — 142.  Measurement  of 
wavelengths.  Crystal  structure. — 143.  X-ray  spectra. — 144.  The  K  and 
L  series. — 145.  Quantum  theory  applied  to  X-rays. — 146.  Secondary 
X-rays.  Absorption.— 147.  Total  range  of  ether  waves. 

138.  Production  of  X-rays. — As  soon  as  the  discovery  of 
X-rays  was  announced  by  Roentgen  in  1897,  speculation  began 
as  to  whether,  like  light,  they  consisted  of  ether- waves;  but  all 
the  evidence  acquired  for  a  long  time  was  conflicting  on  this 
point.  Before  discussing  it,  a  summary  of  what  was  known 
of  the  properties  of  the  rays  a  few  years  ago  will  be  given. 

They  arise  when  an  electric  discharge  is  sent  through  a 
glass  tube  which  has  been  exhausted  to  a  very  low  pressure, 
or  as  it  is  sometimes  stated,  to  a  "high  vacuum. "  At  such 
pressures  the  discharge  in  a  gas  is  carried  largely  by  the 
"cathode  rays,"  which  had  been  'shown  before  Roentgen's  dis- 
covery to  consist  of  a  stream  of  electrons  shot  out  with  very 
high  velocity  from  the  cathode,  or  negative  terminal  of  the 
tube.  The  green  color,  like  a  fluorescent  glow,  which  is  seen 
during  the  discharge,  is  caused  by  the  impact  of  the  cathode- 
ray  electrons  upon  the  glass  walls  of  the  tube.  It  was  soon 
found  that  the  particular  place  where  the  electrons  strike  is 
the  source  of  the  X-rays,  and  the  tubes  are  arranged  so  that 
they  impinge  upon  a  sheet  of  metal  called  the  "anticathode," 
or  "target,"  which  may  or  may  not  be  metalically  connected 
with  the  positive  terminal  of  the  tube.  The  X-rays  then  stream 

out  in  all  directions  from  the 
anticathode  as  shown  in  figure 
131,  where  K  is  the  cathode. 
T  the  anticathode,  A  the 
anode.  The  student  is  warned 
Figure  131  against  thinking  that  X-rays 

are  reflected  cathode  rays.  This  cannot  be,  since  cathode  rays 
are  negatively  charged  particles,  and  X-rays  are  something  en- 
tirely different,  whatever  they  may  be. 

(262) 


PROPERTIES  OF  X-RAYS  263 

139.  Their  properties. — From  the  standpoint  of  the  gen- 
eral public,  the  most  striking  characteristic  of  the  rays  is  their 
ability  to  pass  readily  through  flesh,  somewhat  less  freely 
tli  rough  bone,  and  to  some  extent  even  through  metals,  all  of 
which  are  opaque  to  light.  But  to  a  person  of  analytical  mind 
this  fact  should  be  no  more  surprising  than  that  light  will  pass 
through  glass.  Glass  is  transparent  to  light  and  flesh  is  trans- 
parent to  X-rays,  and  one  of  these  facts  calls  for  explanation 
exactly  as  much  as  the  other. 

A  characteristic  that  is  more  interesting  to  the  physicist 
is  the  ionising  power  of  the  rays.  Any  gas  through  which  they 
pass  becomes  for  a  short  while  a  relatively  good  conductor  of 
electricity,  showing  that  the  passage  of  the  rays  causes  many 
molecules  to  separate  into  positive  and  negative  ions. 

The  rays  affect  a  photographic  plate  very  much  as  light 
does,  and  they  cause  certain  mineral  salts  to  fluoresce  strongly, 
though  they  themselves  do  not  stimulate  vision  in  the  eye. 

X-rays  are  not  subject  to  refraction,  but  pass  straight 
through  a  prism  or  lens  without  any  bending.  Neither  are  they 
regularly  reflected  from  a  polished  surface  as  light  is.  How- 
ever, they  are  to  some  extent  scattered  or  diffused  in  going 
through  matter,  very  much  as  light  is  diffused  in  going 
through  an  attenuated  fog.  The  scattering  occurs  not  only  at 
the  surface,  but  also  throughout  the  material,  even  when  the 
latter  is  a  dense  solid. 

A  grating  has  no  effect  upon  them.  They  pass  right 
through  it  as  they  would  through  any  material,  with  some  scat- 
tering and  absorption.  An  attempt  was  made  to  produce  dif- 
fraction by  passing  X-rays  through  a  very  narrow  aperture, 
or  slit,  the  jaws  of  which  were  made  of  lead,  which  is  very 
opaque.  Under  similar  treatment  light,  as  we  have  seen  in 
section  72,  spreads  out  after  passing  the  aperture,  and  forms 
a  set  of  bright  and  dark  bands,  whose  distance  apart  depends 
upon  the  width  of  the  aperture  and  the  wavelength  of  the 
light.  The  narrower  the  aperture  and  the  longer  the  wave- 
length, the  wider  apart  are  the  bands.  It  was  hoped  by  this 
means  to  show  that  X-rays  also  consist  of  waves,  and  to  meas- 
ure their  length,  but  it  was  found  that  if  they  spread  out  at 
all  it  was  to  such  a  slight  extent  that  the  wavelength,  if  there 


264  LIGHT 

is  any  such  thing,  must  be  something  like  1/5000  that  of  yellow 
light,  or  less. 

X-rays  are  not  all  alike,  for  some  will  penetrate  a  material 
like  aluminum  far  more  readily  than  others.  Rays  of  great 
penetrating  power  are  said  to  be  hard,  those  of  slight  penetrat- 
ing power  soft.  The  hardness  of  the  rays  depends  mainly  upon 
two  factors,  the  material  of  the  anticathode  and  the  degree  of 
exhaustion  of  the  bulb. 

"When  a  stream  of  X-rays  from  a  vacuum-tube  like  figure  131 
falls  upon  a  metal,  under  certain  circumstances  the  latter  gives 
off  electrons  and  also  new  X-rays,  different  from  the  scattered 
rays,  called  secondary  X-rays.  It  is  very  remarkable  that  the 
velocity  with  which  these  electrons  are  sent  off  is  nearly  the 
same  as  that  of  the  electrons  of  the  cathode  ray  stream  which 
produced  the  original  X-rays,  and  is  entirely  independent  of 
whether  the  incident  rays  be  strong  or  weak. 

After  the  discovery  of  radioactive  substances,  with  their 
three  types  of  rays  known  as  a,  /?,  and  y,  it  was  soon  found 
that  the  y-rays  are  similar  in  all  respects  to  X-rays,  except 
that  they  are  more  penetrating  than  the  hardest  rays  obtained 
from  a  vacuum-tube. 

Professor  Marx,  of  Leipsic,  carried  out  a  very  ingenious 
experiment  by  which  he  claimed  to  prove  that  X-rays  travel 
with  the  same  velocity  as  light ;  but  his  method  was  necessarily 
very  complicated,  and  he  was  not  able  to  convince  his  fellow 
investigators  that  his  conclusion  was  justified.  No  accepted 
measurement  of  the  velocity  of  X-rays  has  yet  been  made. 

140,  Are  X-rays  ether  waves? — The  fact  that  X-rays  are 
not  refracted  does  not  necessarily  prove  that  they  are  not  ether- 
waves,  for  it  is  conceivable  that  exceedingly  short  waves  would 
travel  through  matter  with  the  same  velocity  as  through  empt}^ 
space.  Neither  is  the  failure  of  regular  reflection  an  objection, 
when  we  know  that  the  distance  between  the  atoms  of  a  solid 
body  is  something  of  the  general  order  of  10'8  to  10~7  cm., 
that  the  wavelength  of  visible  light  is  in  the  neighborhood  of 
5  X  10"5,  and  that  if  X-rays  have  a  wavelength  it  is  probably 
less  than  10'8.  A  long  string  of  atoms  could  lie  within  a  single 
wavelength  of  yellow  light,  while  on  the  other  hand  a  number 
of  wavelengths  of  X-rays  might  lie  between  two  adjacent  atoms, 
A  surface  that  we  may  regard  as  finely  polished  for  visible 


REFLECTION  OF  X-RAYS  265 

light,  would  therefore  be  an  exceedingly  rough  structure  for 
the  very  short  waves.  In  the  discussion  of  reflection  in  chapter 
IV,  it  was  stated  that  each  point  in  the  straight  line  MN  of 
figure  23  representing  the  reflecting  surface  becomes  the  center 
for  a  secondary  wavelet,  but  it  would  no  doubt  have  been  more 
in  accordance  with  facts  if  it  had  been  said  that  each  atom  of 
the  surface  became  such  a  center.  Figure  132  represents  what 


Figure   132 

a  reflecting  surface  might  look  like  if  it  were  magnified  till 
individual  atoms  became  clearly  perceptible.  The  continuous 
straight  line  MN  of  figure  23  has  become  a  somewhat  irregular 
row  of  atoms  forming  the  top  layer  of  a  body  composed  of 
many  such  atoms.  When  the  wavefront  "WF  strikes  any  one 
atom  it  becomes  a  center  for  a  secondary  wavelet.  1,  2,  3,  4 
and  5  are  the  respective  wavelets  from  the  atoms  a,  b,  c,  d,  and 
e.  Since  these  atoms  are  not  on  a  straight  line,  the  secondary 
wavelets  do  not  lie  tangent  to  a  straight  line,  but  if  the  wave- 
length of  the  light  is  as  long  as  AB  the  amount  by  which  each 
wavelet  falls  off  a  common  tangent  such  as  the  dotted  line  is 
a  very  small  fraction  of  a  wavelength,  and  therefore  the  second- 
ary waves  are  nearly  in  phase  along  this  line.  Therefore,  for 
visible  light  we  are  justified  in  treating  the  polished  surface 
of  a  solid  as  if  it  were  a  continuous  surface.  But  if  the  wave- 


266  LIGHT 

length  were  as  short  as  A'B'  the  conditions  would  be  very 
different,  for  then  any  one  wavelet  might  miss  the  hypothetical 
reflected  wavefront  by  half  a  wavelength  or  more.  Therefore 
there  would  be  no  general  agreement  in  phase  of  the  secondary 
wavelets  along  any  line  which  could  act  as  a  reflected  wave- 
front;  on  the  contrary  the  secondary  wavelets  would  have  a 
general  tendency  to  annul  one  another's  effects,  leaving  only 
a  weak  resulting  effect  in  any  direction,  which  might  account 
for  the  diffuse  scattering  of  X-rays. 

141.    Crystal  reflection  of  X-rays. — But  there  is  one  cir- 
cumstance under  which  we  might  expect  a  sort  of  regular  re- 


Figure  133 

flection  even  for  exceedingly  short  waves,  namely,  when  the 
atoms  are  arranged  in  regular  plane  layers,  as  in  a  crystal. 
This  idea  occurred  to  Professor  Laue,  of  Zurich,  and  at  his 
suggestion  Friedrich  and  Knipping  tried  the  effect  of  several 
different  crystals  in  the  path  of  a  beam  of  X-rays.  They  found 
that  reflection  does  occur,  not  necessarily  from  the  surface  ot 
the  crystal,  but  from  every  plane  within  the  latter  along  which 
atoms  are  distributed  in  a  regular  pattern. 

Laue's  explanation  of  the  phenomena  observed  by  Fried- 
rich  and  Knipping  is  unnecessarily  involved,  and  is  in  fact 
less  useful  than  the  following  method,  which  is  due  to  Messrs. 
W.  H.  and  W.  L.  Bragg,  who  have  done  a  great  deal  of  ex- 
perimental work  with  X-rays.  Their  book,  entitled  "X-rays 
and  Crystal  Structure,"  gives  the  best  account  of  the  subject 
yet  published. 


REFLECTION  FROM  CRYSTALS         267 

In  figure  133  let  AB,  A'B',  A"B",  etc.,  represent  the  rows, 
or  rather  the  contours  of  the  planes,  in  which  atoms  are  regu- 
larly arranged.  Let  PQ  be  a  ray  of  incident  waves.  When 
the  wavefront  strikes  each  point  in  the  layer  AB,  secondary 
wavelets  will  be  sent  out,  and  since  the  atoms  lie  in  a  plane, 
the  secondary  wavelets  will  all  be  tangent  to  a  plane,  and  there 
wili  be  a  reflected  wavefront  for  which  the  ray  QR  is  drawn, 
the  angles  of  incidence  and  reflection  being  equal.  But  this  is 
not  all  for  the  incident  waves  will  pass  on  and  strike  the  next 
layer  of  atoms  A'B',  which  will  also  give  rise  to  a  reflected 
wavefront  represented  by  the  ray  Q'R'.  In  the  same  way  one 
layer  after  another  would  take  up  the  act  of  reflection,  until, 
after  penetrating  to  a  sufficient  depth  within  the  crystal,  the 
incident  waves  are  too  much  weakened  to  produce  sensible  re- 
flection. Here  we  have  to  do  with  reflection  from  many  parallel 
surfaces,  spaced  at  equal  intervals.  The  condition  is  really 
quite  analogous  to  the  reflection  of  light  from  a  material  film 
of  uniform  thickness  (see  section  76  and  figure  80),  although 
there  are  certain  important  differences.  Here  we  have  to  do 
with  a  series  of  reflected  beams  produced  by  reflection  from 
many  different  planes,  rather  than  between  only  two  planes. 
All  the  reflections  in  the  present  case  are  of  the  same  kind,  so 
that  there  is  no  occasion  for  a  difference  in  phase  due  to  re- 
flection alone.  Moreover,  the  index  of  refraction  is  to  be  re- 
garded as  having  the  value  1,  and  the  angles  of  incidence  and 
refraction  are  equal.  Bearing  these  facts  in  mind,  the  differ- 
ence in  path  between  rays  PQR  and  PQ'R',  or  between  PQ'R' 
and  PQ"R",  etc.,  can  be  gotten  from  the  analogous  case  of 
figure  80.  It  is  simply 

2t.  cos  r,  or  2t.  cos  i. 

t  being  the  perpendicular  distance  between  successive  layers 
of  atoms.  X-ray  workers  have  found  it  more  convenient  to 
express  their  results  in  terms  of  what  they  call  the  "glancing- 
angle,"  a,  than  in  terms  of  i,  the  angle  of  incidence,  and  there- 
fore the  difference  in  path  is  usually  written 

2t.  sin  a 

If  this  quantity  is  equal  to  a  wavelength,  or  to  any  whole  num- 
ber of  wavelengths,  the  reflections  from  the  different  layers 


268  LIGHT 

will  reinforce  one  another,  and  the  reflection  will  be  strong. 
The  formula  for  strong  reflection 

2t.  sin  a  =  nA 

reminds  one  of  the  formula  for  the  grating,  and  indeed  a 
crystal  does  provide  a  natural  grating  for  exceedingly  short 
wavelengths;  but  it  differs  from  the  ordinary  grating  in  that 
the  regular  spacing  extends  in  three  different  directions  instead 
of  only  in  one,  and  this  fact  makes  important  differences.  With 
the  ordinary  grating,  the  angle  of  incidence  of  the  light  may 
be  zero,  and  we  look  for  various  wavelengths  at  various  angles ; 
but  with  the  three-dimensional  grating,  or  "space-lattice,"  the 
angle  of  incidence  as  well  as  the  angle  of  reflection  must  have 
a  particular  value,  and  only  one  wavelength  can  be  shown  at  a 
time. 

142.  Measurement  of  wavelengths.  Crystal  structure. — 
The  procedure  in  investigating  the  wavelengths  of  X-rays  is, 
in  crude  outline,  as  follows.  The  X-ray  tube,  such  as  that 
shown  in  figure  131,  is  enclosed  in  a  lead  box  with  a  narrow 
slit  at  a  certain  place,  so  that  only  a  very  fine  beam  of  the  rays 
gets  out  through  the  slit.  A  crystal  is  held  in  the  path  of  this 
beam,  and  slowly  turned  so  that  the  angle  a  is  changed.  At 
the  same  time,  a  photographic  plate,  or  a  device  for  measuring 
the  ionization  produced  by  the  rays,  is  turned  about  the  same 
axis  at  twice  the  rate  of  motion,  so  as  to  always  be  in  position 
to  receive  the  reflected  rays  if  there  are  any.  At  certain  posi- 
tions a  blackening  of  the  plate  or  a  functioning  of  the  ionization 
apparatus  shows  a  reflection,  and  the  angle  a  is  measured.  In 
addition  to  this  angle  a  we  must  of  course  know  the  distance  t 
between  the  layers  of  atoms.  The  man- 
ner in  which  this  is  found  can  be  best 
understood  from  the  following  exam- 
ple, the  special  crystal  being  rocksalt. 
It  is  composed  of  the  atoms  of  sodium 
and  chlorine  in  equal  numbers  (formu- 
la NaCl)  and  its  density  is  2.17.  From 
considerations  in  regard  to  its  crys- 
tal form,  which  are  well  supported  by  the  experiments  with 
X-rays,  we  are  convinced  that  the  atoms  in  this  crystal  are 
arranged  in  a  cubical  pattern  as  illustrated  in  figure  134,  where 


6-- 


.rV 


4 


Figure    134 


MEASUREMENT  OF  WAVELENGTHS     269 

the  white  circles  represent  sodium  atoms,  the  black  chlorine 
atoms.  To  each  small  cube  of  the  structure  one  atom  must  be 
assigned,  as  can  be  readily  understood  if  the  reader  notes  that 
cubes  of  the  same  size  can  be  drawn  with  an  atom  at  the  cen- 
ter of  each,  the  cubes  filling  the  entire  space.  Therefore,  since 
t  is  the  length  of  each  side  of  a  little  cube,  each  atom  commands 
a  space  of  volume  t3,  and  in  one  cubic  centimeter  of  crystal 
there  are  1/t3  atoms.  Since  one  cubic  centimeter  has  a  mass 
of  2.17  grams,  the  average  mass  of  the  atoms  must  be 

2.17  ¥  1/t3  =  2.17t3 

Now  the  atomic  weight  of  sodium  is  approximately  23,  and  that 
of  chlorine  35,  so  that  the  average  atomic  weight  is  29,  that 
is  the  average  mass  of  the  atoms  making  up  the  crystal  is  29 
times  that  of  the  hydrogen  atom.  From  a  number  of  physical 
and  chemical  lines  of  attack,  we  know  the  mass  of  the  hydrogen 
atom  to  be  1.64  X  10'24,  therefore  the  average  atom  of  rock- 
salt  has  mass 

29  X  1-64  X  1C'24  —  47.56  X  10  24 
We  can  then  put 

2.17  t3  =  47.56  X  10-24 
t  =  2.80  X  lO'8 

One  might  be  inclined  to  wonder  whether  it  is  not  a  mole- 
cule, rather  than  an  atom,  which  is  located  at  each  corner  of 
a  cube  of  the  crystal  lattice.  The  way  in  which  it  was  proved 
that  it  is  a  matter  of  atoms  rather  than  molecules  will  not  be 
taken  up  here.  It  is  fully  explained  in  Bragg 's  book,  from 
which  most  of  the  facts  of  this  chapter  are  taken. 

Having  the  value  of  t,  it  is  possible  to  find  the  wavelength 
of  the  X-rays  emitted  by  any  anticathode.  A  palladium  anti- 
cathode  is  found  to  emit  rays  of  wavelength  .576  X  10~8  as  well 
as  certain  other  wavelengths.  One  wavelength  being  known, 
it  is  now  possible  to  reverse  the  procedure  outlined  above,  and 
find  the  structure  of  crystals  less  simple  than  rocksalt.  Thus 
we  are  provided  with  a  new  method,  by  which  we  can  determine 
the  inner  structure  of  crystals  as  well  as  measure  the  wave- 
lengths of  X-rays,  although  it  is  only  in  the  latter  that  we  are 
at  present  interested. 


270  LIGHT 

143.  X-ray  spectra. — We  carry  over  into  the  discussion  of 
X-rays  a  number  of  the  terms  familiar  in  connection  with  visi- 
ble light.     Thus  we  speak  of  the  X-ray  "spectrum"  of  a  ma- 
terial,  meaning  the  total  array  of  wavelengths  which  it  emits 
when  acting  as  the  anticathode  of  an  X-ray  tube.     When  a 
spectrum  extends  over  a  considerable    range    of    wavelengths, 
without  the  absence  of  any  wavelength  within  that  range,  we 
say  it  is  "continuous."     On  the  other  hand,  a  spectrum  con- 
sisting of  several  distinct  wavelengths  only  is  said  to  be  made 
up  of  so  many  "lines."     Any  metal    serving    as    anticathode 
emits  a  continuous  spectrum,  but  also,  if  the  vacuum  be  good 
enough  and  the  consequent  speed  of    the    cathode    rays    great 
enough,  certain  characteristic  lines,  much  more  intense  than  the 
continuous  spectrum.     Each  element  emits    a    group    of    lines 
similar  to  those  characteristic  of  other  elements,  but  the  higher 
the  atomic  weight  the  shorter  the  wavelengths  in  the  group. 
If  we  pick  out  corresponding  lines  for  different  elements,  cal- 
culate their  frequencies  from  the  general    relation    that    fre- 
quency equals  velocity  of  light    divided    by    wavelength,    and 
then  plot  the  square-roots   of  these    frequencies    as    abscissa} 
against  the   "atomic   numbers"*   as   ordinates,   the   points   lie 
very  accurately  upon  a  curve  which  is  nearly  straight,   that 
is,  the  square  roots  of  the  frequencies  are  proportional  to  the 
atomic  numbers. 

144.  The  K  and  L  series. — Certain  elements  of  neither  very 
high  nor  very  low  atomic  number  yield  two  groups  of  lines, 
the  K  series  and  the  L  series,  of  which  the  former  is  consti- 
tuted of  shorter  wavelengths.     For  elements  .of  lower  atomic 
numbers  only  the  K  series  has  been  found,  for  members  of  higher 
numbers  only  the  L,  series.     The  rule  of  proportionality  be- 
tween square-root  of  frequency  and  atomic  number  holds  for 
each  series  separately.     If  this  rule  be  carried    down    to    the 
case  of  hydrogen,  for  which  naturally   X-rays  cannot  be  ob- 
tained in  the  usual  way,  it  indicates  that  the  K  series  for  hydro- 

*The  atomic  number  is  the  ordinal  number  of  the  element  when 
the  whole  list  of  elements  is  arranged  in  the  order  of  increasing  atomic 
weight.  For  elements  of  smaller  atomic  weights,  it  is  nearly  equal  to 
half  the  atomic  weight.  Recent  investigations  show  that  the  atomic 
number  is  far  more  indicative  of  an  element's  physical  and  chemical 
properties  than  the  atomic  weight. 


QUANTUM  THEORY  AND  X-RAYS  271 

gen  should  come  in  what  we  have  hitherto  regarded  as  the  ex- 
treme ultraviolet  region,  where  part  of  a  series  of  lines  has 
actually  been  discovered  by  Lyman;  also,  that  the  L  series 
should  about  correspond  to  the  hydrogen  series  in  the  visible 
spectrum  shown  in  figure  62,  for  which  Balmer's  series-formula 
applies.  That  is,  hydrogen  being  the  lightest  known  element, 
its  X-ray  spectrum  seems  to  be  composed  of  wavelengths  so 
long  as  to  bring  the  lines  in  the  visible  and  the  extreme  ultra- 
violet regions.  Possibly  the  ordinary  visible  series  spectra  of 
the  heavier  elements  could  legitimately  be  regarded  as  repre- 
senting X-ray  series  of  longer  wavelength  than  the  L-series. 

145.  Quantum  theory  applied  to   X-rays.— Mention  has 
been  made  in  section  59  of  the  quantum  theory^  introduced  by 
Planck  to  explain  the  radiation  of  an  absolutely  black  body. 
There  appears  in  this  theory  a  constant  h,  numerically  equal 
to  6.55  X   10'27,  whose  significance  is  that  a  radiating  center 
which  sends  out  light  of  frequency  v  can  do  so  only  in  energy- 
amounts  equal  to  a  multiple  of  hv.    Curiously  enough,  this  same 
constant  appears  in  X-ray  phenomena,  for  it  is  found  that  an 
X-ray  tube  can  emit  the  radiations  characteristic  of  its  anti- 
cathode  metal,  of  frequency  v,  only  when  the  velocity  of  the 
electrons  constituting  the  cathode-ray  stream  is  great  enough  so 
that  their  energy  is  at  least  equal  to  hv.    Of  course  we  have  no 
rational  explanation  for  this  fact,  or  for  any  other  case  in  which 
the  constant  h  appears. 

146.  Secondary  X-rays.  Absorption. — It  was  stated  earlier 
in  this  chapter  that  characteristic  secondary  X-rays  are  pro- 
duced under  certain  circumstances  when  primary  rays  from  a 
vacuum-tube  fall  upon  a  metal  outside  the  tube.     These  sec- 
ondary rays  have  the  wavelengths  of  the  K  or  the  L  series  ap- 
propriate to  the  metal  which  emits  them.     These  rays  are  pro- 
duced only  when  the  primary  rays  have  a  somewhat  higher 
frequency   (shorter  wavelength). 

The  absorption  of  X-rays  by  a  given  material  is  different 
according  to  the  wavelength  of  the  rays,  just  as  we  should  ex- 
pect. But  the  wavelength  most  strongly  absorbed  is  not  the  same 
as  that  which  the  material  could  under  proper  stimulus  emit,  as 
we  should  infer  from  analogy  with  phenomena  in  the  visible 
region  (absorption  by  sodium  vapor,  or  absorption  by  the  sun's 
atmospheric  gases  of  those  wavelengths  which  they  can  emit). 


272  LIGHT 

For  X-rays,  the  emitted  wavelengths  are  always  longer  than 
those  most  readily  absorbed,  as  in  the  phenomenon  of  fluores- 
cence. 

147.  Total  range  of  ether  waves. — The  facts  related  in 
this  chapter  show  beyond  reasonable  doubt  that  X-rays  are 
ether- waves  of  the  same  general  character  as  visible  light, 
though  of  exceedingly  short  length.  They  may  be  regarded  as 
ultraviolet  carried  to  the  extreme,  though  the  term  ultraviolet 
is  usually  meant  to  include  only  waves  stimulated  by  the  same 
general  methods  used  to  produce  visible  light,  such  as  high 
temperature,  ordinary  electric  spark  discharges  between  metal 
points  close  together,  discharge  through  gases  at  pressures  not 
excessively  low,  etc.  There  is  still  a  gap  between  the  shortest 
ultraviolet  waves  produced  by  such  means  and  the  longest 
X-rays  produced  either  directly  or  indirectly  by  cathode-ray 
discharge,  but  we  are  now  acquainted  with  an  extraordinarily 
large  range  of  ether  waves,  from  the  exceedingly  short  y-ray 
waves  emitted  by  radioactive  substances  at  one  end  of  the  scale 
to  the  waves  familiar  under  the  head  of  wireless  telegraphy  at 
the  other;  and  such  gaps  as  exist  in  the  range  are  relatively 
small.  The  following  table  gives  the  salient  points  in  the  range 
of  ether- waves,  according  to  length,  as  we  now  know  them: 

wavelength,  in  cms. 

Shortest  known  waves   ( y-ray s  from  radium)  10'* 

Longest  X-rays  from  cathode-ray  discharge  1.2  X  10"7 

Shortest  ultraviolet  from  spark  discharge  2 .7  X  10"6 

Shortest  visible  3.5  X  10"r> 

Longest  visible  7.0  X  10'5 

Longest  infrared   (from  mercury-arc  lamp)  3.4  X  10  2 

Shortest  waves  from  electrical  oscillations  in 

constructed  apparatus  2.0  X  10'1 

Approximate  length  used  in  wireless  105 

Longest  waves  attainable  no  limit. 


CHAPTER  XVII. 

148.  Review  of  the  development  of  light-theory. — 149.  Lines  of 
modern  investigation.  150.  The  Zeeinan  effect. — 151.  Lorentz's  theory. 
152.  The  Stark  effect. — 153.  The  photo-electric  effect. — 154.  Atom-models. 
— 155.  Bohr's  theory  of  the  hydrogen  atom. 

148.  Review  of  the  development  of  light-theory. — The  de- 
velopment of  the  theory  of  light  given  in  the  preceding  chap- 
ters follows  roughly  the  chronological  order  of  the  history  of 
the  subject.  It  is  of  interest  to  notice  that  there  are  a  number 
of  stages  in  the  development,  with  rather  clearly  marked  divid- 
ing points. 

Let  us  turn  our  attention  first  to  the  old  corpuscular 
theory.  It  could  never  have  held  its  place  so  long,  but  for  the 
reverence  in  which  the  authority  of  Sir  Isaac  Newton,  its  chief 
advocate,  was  held,  long  after  his  death,  particularly  in  Eng- 
land. Considerable  progress  was  made  under  this  theory,  par- 
ticularly in  regard  to  colors  and  the  phenomena  of  reflection 
and  refraction. 

Nevertheless,  the  final  adoption  of  the  wave  theory  was  a 
distinct  break,  for  it  offered  satisfactory  explanations  of  such 
phenomena  as  interference  and  diffraction,  which  were  serious 
stumbling-blocks  to  the  corpuscular  theory.  Also,  it  explained 
reflection,  refraction,  and  color  differences  in  a  more  rational 
manner,  brought  about  the  invention  of  gratings  and  inter- 
ferometers, and  thus  led  to  the  detailed  and  accurate  study  of 
spectra. 

The  discovery  of  polarization  may  be  said  to  have  intro- 
duced a  third  stage,  for  it  proved  that  light  waves,  of  whose 
nature  nothing  had  previously  been  known  except  their  length, 
were  not  longitudinal  but  transverse.  A  period  of  active  specu- 
lation as  to  the  nature  of  the  ether  and  ether  waves  followed, 
leading  to  the  "elastic-solid"  theories  mentioned  in  chapter 
XIV.  These  very  ably  worked  out  theories  are  excellent  exam- 
ples of  the  application  of  mathematical  analysis  to  physical 
phenomena. 

Meanwhile,  the  mathematical  theory  of  electrical  phenom- 
ena was  also  being  perfected,  and  this,  as  we  have  seen  in 

(273) 


274  LIGHT 

chapter  XIV,  enabled  Maxwell  to  predict  the  existence  of  elec- 
tromagnetic waves  and  announce  the  electromagnetic  theory  of 
light,  which  fully  supplanted  the  older  theories  after  the  ex- 
periments of  Hertz.  This  brought  the  subject  to  the  fourth, 
and  so  far  the  final,  stage. 

It  must  be  admitted  that,  at  the  present  time,  the  electro- 
magnetic ether-wave  theory  of  light  does  not  stand  on  an 
absolutely  secure  foundation,  for  certain  experimental  facts 
cast  some  doubt  upon  it, — namely,  those  facts  that  gave  rise 
to  the  relativity  theory  and  the  quantum  theory.  Strange  as 
it  may  seem,  the  old  corpuscular  theory  of  light  could  account 
very  nicely  indeed  for  the  Michelson-Morley  experiment  (chap- 
ter IX)  and  for  some  of  those  phenomena  in  which  the  constant 
h  of  the  quantum  theory  appears.  In  fact,  one  might  regard 
the  relativity  theory  and  the  quantum  theory  as  hypotheses  in- 
troduced to  make  the  wave  theory  square  with  these  facts,  and 
the  necessity  for  such  additional  hypotheses  may  be  regarded  as 
a  weakness  in  the  structure.  On  the  other  hand,  the  corpuscu- 
lar theory  is  condemned  by  interference,  diffraction,  and  polari- 
zation, which  seem  impossible  to  explain  except  in  terms  of 
waves. 

149.  Lines  of  modern  investigation. — Since  the  introduc- 
tion of  the  electromagnetic-theory,  progress  has  been  very 
rapid,  and  has  followed  several  general  lines.  First,  our  knowl- 
edge of  the  range  of  ether  waves  has  been  greatly  extended, 
not  only  by  thrusting  farther  out  into  the  ultraviolet  and  infra- 
red, but  also  by  an  abrupt  jump  into  the  region  of  very  short 
waves  (X-rays)  and  another  into  that  of  very  long  waves 
(Hertz  waves). 

Another  line  of  attack  has  been  the  study  of  the  radiation 
from  an  absolutely  black  body  (see  chapter  VII).  This  has 
involved  a  great  deal  of  very  careful  experimental  work,  as 
well  as  difficult  theoretical  study,  leading,  as  we  have  seen,  to 
the  formulation  of  the  quantum  theory. 

Third,  a  number  of  previously  unknown  phenomena  have 
been  discovered  and  investigated,  partly  optical  and  partly 
electrical  in  nature.  These  have  to  do  with  the  emission  and 
absorption  of  light,  that  is  with  the  relation  between  radiated 
energy  and  matter,  and  they  are  interpreted  in  terms  of  the 
electron  theory  of  matter.  The  most  striking  of  these  phenomena 


ZEEMAN  EFFECT 


275 


is  X-rays.  Since  this  subject  has  been  dealt  with  at  some  length 
as  the  special  subject  of  chapter  XVI,  we  shall  not  discuss  it 
further  here,  but  proceed  to  a  brief  discussion  of  a  few  of  the 
others. 

150.  The  Zeeman  effect. — Long  before  Maxwell  formulated 
the  electromagnetic  theory  of  light,  Michael  Faraday,  who  had 
discovered  that  a  beam  of  light  passing  through  a  magnetic 
field  has  its  plane  of  polarization  rotated,  conceived  the  idea 
of  placing  a  source  of  light  directly  in  the  magnetic  field,  to 
see  whether  the  latter  produced  any  effect  upon  the  spectrum 
lines.  Although  we  now  know  there  is  such  an  effect,  Faraday 
was  unable  to.  detect  it,  because  the  spectroscopes  then  available 
were  not  efficient  enough.  Rowland  also  tried  the  experiment 
Avithout  success.  In  1897  it  was  tried  again  by  Zeeman,  using  a 
strong  magnet  and  a  good  grating  spectroscope.  The  effect 
which  was  discovered  has  been  named  the  '  *  Zeeman  effect. ' ' 

In  figure  135  let 
MINI  be  the  two  parts 
of  a  strong  electro- 
magnet, each  termin- 
ating in  a  conical 
pole-piece,  leaving  a 
small  space  between, 
where  the  magnetic 
field  is  very  strong. 
The  source  of  light, 
S,  is  placed  in  this 
strong  field.  The  slit 
of  the  spectroscope 
may  be  placed  either 

at  sr  (broadside  position),  or  at  S2  (end  position).  In  the  latter 
case  a  hole  must  be  bored  through  one  pole-piece  and  magnet 
core,  to  let  the  light  through.  In  either  case,  an  image  of  the 
source  is  focussed  upon  the  slit  by  means  of  a  lens.  The  results 
found  in  the  earlier  experiments  of  Zeeman,  and  of  others  who 
took  up  this  study,  may  be  stated  as  follows:  In  the  broadside 
position,  each  spectrum  line  is  changed  by  the  presence  of  the 
magnetic  field  into  a  group  of  three  lines  very  close  together 
(triplet).  The  middle  line  of  the  triplet  is  polarized  in  a  plane 
perpendicular  to  the  plane  of  the  figure,  and  the  two  outer 


~ 


Figure    135 


276  LIGHT 

lines  are  polarized  in  the  plane  of  the  figure.  In  the  end  posi- 
tion, the  magnetic  field  causes  each  spectrum  line  to  become  a 
close  doublet,  consisting  of  two  lines,  one  of  which  is  circularly 
polarized  in  the  right-hand  direction,  the  other  in  the  left-hand 
direction. 

151.  Lorentz's  theory. — These  results  were  given  a  very 
clear  explanation  in  terms  of  the  electron  theory  by  H.  A. 
Lorentz.  He  supposed  the  light  to  be  sent  out  from  an  atom  by 
the  vibrations  of  electrons  within  the  atom.  Each  electron  was 
conceived  to  be  bound  to  the  center  of  its  path  by  a  sort  of  elastic 
force,  that  is  a  force  that  varies  directly  as  the  distance.  Suppose 

OX,  OY,  and  OZ,  figure  136,  to 
be  three  directions  perpendicular 
to  one  another.  We  take  one  of 
these,  OX,  to  represent  the  direc- 
tion of  the  magnetic  field,  and  sup- 
pose 0  to  be  the  center  about  which 
an  electron  vibrates.  The  direction 
Figure  136  of  vibration  might  be  anything, 

so  we  choose  a  direction  OR  at  random,  the  length  OR  repre- 
senting the  amplitude  of  the  vibration.  "We  want  to  find 
how  such  a  vibration  would  be  affected  by  the  presence 
of  a  magnetic  field  in  the  direction  OX.  To  do  this,  it 
is  best  to  resolve  the  vibration  into  certain  components. 
In  the  first  place,  a  vibration  of  amplitude  OR  is  equi- 
valent to  one  of  amplitude  OP  parallel  to  OX  and  one  of 
amplitude  OQ  in  the  plane  of  OY  and  OZ.  (OQ  and  OR  lie 
in  a  plane  through  OX  perpendicular  to  the  plane  of  OY  and 
OZ,  and  the  figure  OPRQ  is  a  rectangle.)  The  vibration  of 
amplitude  OQ  can  again  be  resolved  into  a  right-handed  and 
a  left-handed  circular  motion  about  OX  as  axis,  as  was  done 
by  Fresnel  in  explaining  the  rotation  of  the  plane  of  polariza- 
tion by  quartz.  (See  figure  120,  chapter  XIII,)  Thus  we  may 
think  of  the  single  electron  as  replaced  by  three  different  elec- 
trons, of  which  one  vibrates  back  and  forth  in  the  direction 
OX,  one  rotates  about  OX  in  one  direction,  and  the  third  in 
the  opposite  direction.  Since  an  electron  is  a  negatively 
charged  body,  a  moving  electron  constitutes  a  current.  A  mag- 
netic field  has  no  effect  upon  a  current  in  the  same  direction 
as  the  field,  and  therefore  the  vibration  along  OX  goes  on  just 


> 

ZEEMAN  EFFECT  277 

as  if  the  field  were  absent.  The  two  rotary  motions,  however, 
constitute  currents  across  the  field,  and  it  is  well  known  that 
the  effect  of  the  field  upon  such  a  current  is  to  exert  a  force  on 
the  latter,  perpendicular  both  to  the  field  and  to  the  current, 
that  is,  either  toward  the  center  0  or  away  from  it,  according 
to  the  direction  of  the  rotation.  That  is,  one  of  the  electrons 
rotating  about  OX  as  axis  will  have  the  elastic  force  pulling 
it  to  the  center  somewhat  strengthened  by  the  addition  of  a 
magnetic  pull  in  the  same  direction.  This  •  electron  will  there- 
fore have  its  angular  speed  increased,  its  period  diminished. 
The  electron  rotating  in  the  opposite  direction  will  have  its 
pull  toward  the  center  somewhat  weakened  by  the  magnetic 
action.  It  will  therefore  rotate  somewhat  slower,  that  is  with 
a  longer  period. 

The  vibration  along  OX  sends  out  waves  of  the  same  peri- 
od, and  the  same  wavelength,  as  the  vibration  in  absence  of 
a  magnetic  field.  These  will  have  their  electric  vibrations 
parallel  to  OX,  so  that  their  plane  of  polarization  will  be  per- 
pendicular to  OX,  i.  e.,  in  the  plane  of  OY  and  OZ.  They 
will  pass  out  in  all  directions  except  the  positive  or  negative 
direction  of  the  X-axis,  the  axis  of  the  magnet,  but  most 
strongly  in  the  YZ  plane,  so  that  light  of  this  wavelength  would 
be  observed  in  the  broadside  position,  but  not  in  the  end  posi- 
tion. The  circular  vibration  would  send  out  circularly  polar- 
ized light  in  both  directions  along  the  X-axis,  and  elliptically 
polarized  light  in  every  other  direction  except  directions  in  the 
YZ  plane.  In  this  plane,  since  the  circles  in  which  the  elec- 
trons rotate  are  seen  edge-on,  plane-polarized  light  would  be 
the  result.  The  direction  of  the  electric  vibrations  would  be 
perpendicular  to  the  magnetic  field,  so  that  the  plane  of 
polarization  would  be  parallel  to  the  field.  The  wavelength 
would  be  slightly  shorter  in  one  case,  and  slightly  longer  in 
the  other,  than  the  natural  wavelength.  Thus,  the  three  differ- 
ent wavelengths  seen  in  the  broadside  position,  and  the  two  in 
the  end  position,  are  fully  accounted  for,  even  to  the  state  of 
polarization.  Moreover,  the  theory  indicates  that  the  change 
in  wavelength  of  the  two  outer  lines  of  the  triplet  depends  upon 
the  ratio  of  the  charge  upon  the  electron  to  its  mass,  and  the 
measured  change  gives  about  the  same  value  to  this  ratio  that 


278  LIGHT 

was  known  to  be  right  from  electrical  experiments  with  cathode 
rays. 

Unfortunately  for  Lorentz's  theory,  it  was  soon  found, 
when  stronger  magnetic  fields  were  available,  that  many  lines 
split  up  into  four,  five,  and  even  more  different  component 
lines  in  the  field.  In  fact,  the  simple  triplet  seems  rather  the 
exception  than  the  rule.  Lorentz's  theory  was  incapable  of 
explaining  these  more  complicated  cases  of  the  Zeeman  effect; 
and  the  only  theory  which  has  been  applied  to  them  with  any 
degree  of  success  is  the  theory  of  Ritz,  which  will  be  mentioned 
further  on  in  this  chapter. 

152.  The  Stark  effect.  A  phenomenon  somewhat  analogous 
to  the  Zeeman  effect  was  discovered  by  Stark.     It  is  a  similar 
splitting  up  of  the  spectrum  lines  when  the  source  of  light  is 
placed  in  a  strong  electrostatic  field.     It  has  not  been  studied 
to  the  same  extent  as  the  Zeeman  effect. 

153.  The  photo-electric  effect. — Hallwachs  discovered  in 
1888  that  a  negatively  charged  zinc  plate,  if  illuminated  with 
ultraviolet  light,  would  lose  its  charge.     This  phenomenon,  the 
photo-electric  effect,   is   also  shown  by:  other  metals,  and  with 
some  even  visible  light  is  effective.'    The  action  of  the  light  is 
to  cause  electrons  to  be  shot  out  from  <the  plate.     The  phenome- 
non  is  difficult  to  work  with,  because  it  is  necessary  for  the 
surface  of  the  metal  to  be  perfectly  clean  and  free  from  tar- 
iiish,  and  most  metals  tarnish  under  the  action  of  the  air  fast 
enough   to   make   the   effectiveness   decrease   very   rapidly.      In 
some  experiments,  the  metal  and  other  parts  of  the  apparatus 
were  enclosed  within  a  highly  exhausted  vacuum-tube,   which 
also  contained  a  device  whereby  a  thin  shaving  could '  be  cut 
off  the  metal  at  any  time,  leaving  a  fresh  clear  surface. 

Suppose  that  the  illuminated  metal,  which  we  shall  call 
A,  is  placed  face  to  face  with  another  metal  plate  B,  which  is 
shielded  from  the  ultraviolet  light,  so  that  B  may  receive  the 
electrons  shot  off  from  A.  If  B  is  kept  at  zero  potential  by 
being  connected  with  the  earth,  while  A  is  kept  at  a  positive 
potential  V,  then,  provided  V  be  high  enough,  the  electrons 
will  be  kept  from  reaching  B  by  the  attraction  of  the  positive 
charge  upon  A,  which  will  draw  them  back.  The  magnitude 
of  the  potential  V  which  is  just  sufficient  to  accomplish  this  is 
very  simply  related  to  the  speed  with  which  the  electrons  are 


ATOM-MODELS  279 

emitted.  If  v  represents  this  speed,  and  m  the  mass  of  an  elec- 
tron, the  kinetic  energy  with  which  they  leave  A  is  mvY2.  If 
e  represents  the  charge  upon  the  electron,  the  expenditure  of 
work  necessary  to  take  it  from  a  potential  V  to  potential  0 
is  equal  to  the  product  of  charge  and  potential-difference,  or 
eV.  If  they  are  just  stopped  before  reaching  B,  the  kinetic 
energy  mv2/2  is  just  lost  in  doing  this  work,  therefore 

eV  =  mv2/2 

Since  the  values  of  m  and  of  e  are  well  known  from  electrical 
experiments,  this  equation  enables  us  to  find  v  by  measuring 
the  potential  V  just  sufficient  to  prevent  any  current  passing 
between  A  and  B,  for  the  transfer  of  electrons  constitutes  a 
current. 

It  has  been  found  by  such  experiments  that  the  velocity 
of  the  emitted  electrons  is  exactly  the  same,  whether  the  beam 
of  ultraviolet  light  be  strong  or  weak,  although  of  course  the 
number  emitted  per  second  is  greater  with  stronger  illumina- 
tion. (Notice  that  a  similar  relation  was  found  to  hold  for  the 
emission  of  electrons  and  secondary  X-rays,  under  the  stimulus 
of  another  beam,  of  X-rays,  chapter  XVI).  The  velocity  does 
depend,  however,  upon  the  wavelength  of  the  light.  The 
shorter  the  wavelength  (i.  e.,  the  higher  the  frequency),  the 
greater  is  the  velocity,  and  the  relation  between  the  electron 
velocity  and  the  frequency  of  the  exciting  light  involves  the 
constant  h  of  the  quantum  theory. 

mv2/2  =  hv  — a 

where  v  is  the  frequency  and  a  is  a  constant  characteristic  of 
the  particular  metal  employed.  The  equation  may  be  inter- 
preted in  the  following  manner:  As  soon  as  an  atom  has  ab- 
sorbed from  the  incident  light  a  quantum  of  energy,  i.  e.,  the 
amount  hv,  an  electron  is  emitted.  Part  of  the  absorbed  energy, 
the  amount  a,  is  used  up  in  getting  free  from  the  metal,  and 
the  remainder,  hv  —  a,  is  retained  by  the  electron  as  kinetio 
energy.  If  the  frequency  is  so  small  (wavelength  so  great) 
that  hv  is  less  than  a,  the  electron  cannot  escape,  and  therefore 
short  wavelengths  are  necessary  for  the  photoelectric  effect. 

154.    Atom-models. — One  of  the  aims  of  all  physical  re- 
search, as  has  already  been  mentioned  in  this  book,  is  to  explain 


280  LIGHT 

the  structure  and  behavior  of  atoms.  If  this  aim  is  ever  to  be 
accomplished,  we  must  pay  due  attention  to  what  information 
the  chemists  have  accumulated,  as!  well  as  to  such  physical 
phenomena  as  spectral  series,  X-rays,  the  Zeeman  effect,  the 
photoelectric  effect,  the  behavior  of  gases  under  electric  dis- 
charge, etc.  It  now  seems  fairly  certain  that  an  atom  is  made 
up  of  a  positive  charge,  the  nucleus,  and  one  or  more  electrons. 
If  the  atoms  are  arranged  in  the  order  of  increasing  atomic 
weights,  hydrogen,  helium,  lithium,  beryllium,  boron,  carbon, 
nitrogen,  oxygen,  etc.,  the  first,  hydrogen,  is  believed  to  have 
a  small  nucleus  and  a  single  electron, — helium  a  nucleus  of 
twice  the  charge  of  the  hydrogen  nucleus,  with  two  electrons, 
—lithium  a  nucleus  of  three  times  the  charge  of  the  hydrogen 
nucleus,  with  three  electrons, — and  so  on,  an  element  of  high 
atomic  weight  having  a  high  nuclear  charge  and  a  correspond- 
ingly large  number  of  electrons.  The  "atomic  number"  of  an 
element  is  the  number  of  its  electrons,  or  the  ratio  of  its  posi- 
tive nuclear  charge  to  that  of  the  hydrogen  atom. 

So  far,  physicists  and  chemists  are  fairly  in  agreement, 
but  they  differ  in  their  ideas  as  to  the  arrangement  of  the 
electrons.  It  is  supposed  that  the  outer  electrons  are  responsi- 
ble for  the  chemical  valence  of  the  element;  and  certain  facts> 
particularly  those  concerning  the  compounds  of  carbon,  which 
has  a  valence  four  (that  is,  a  carbon  can  hold  four  hydrogen 
atoms  in  combination)  seem  to  indicate  that  these  valence-elec- 
trons have  definite  relations  to  one  another  in  space.  For  this: 
reason  chemists  are  inclined  to  regard  them  as  stationary  in  the 
atom,  each  electron  having  a  fixed  position.  On  the  other 
hand,  the  physicists,  having  more  in  mind  that  the  equilibrium 
of  the  electrons  must  be  explained,  look  upon  them  as  being 
in  revolution  about  the  nucleus.  Perhaps  some  means  of  recon- 
ciling these  apparently  diverging  points  of  view  may  be  found. 

We  shall  ignore  any  further  consideration  of  stationary 
electrons,  and  confine  our  attention  to  atom-theories  in  which 
the  electrons  are  in  motion.  Lorentz's  theory  to  account  for 
the  Zeeman  effect,  which  is  really  an  atom-theory,  has  already 
been  mentioned.  It  is  formulated  on  the  assumption  that  an 
electron  is  attracted  toward  the  center  of  its  orbit  by  a  force- 
directly  proportional  to  the  distance.  Now  if  this  force  is  the 
attraction  beween  the  nucleus  and  the  electron,  it  should  vary 


ATOM-MODELS  281 

inversely  as  the  square  of  the  distance,  like  any  other  electro- 
static force,  provided  the  electron  is  outside  the  nucleus.  It 
would  vary  directly  as  the  distance  only  if  the  nucleus  were  in 
the  form  of  a  sphere  of  positive  electricity  with  the  charge 
uniformly  distributed  throughout  its  volume  and  the  electron 
moving  inside  this  sphere.  The  atom  would  then  be  a  globe 
of  positive  electricity  with  enough  electrons  moving  inside  to 
neutralize  the  positive  charge,  each  electron  attracted  to  the 
center  by  a  force  proportional  to  its  distance  from  it. 

There  is  a  vital  difference  between  a  central  force  varying 
as  the  distance  and  one  varying  inversely  as  the  distance 
squared,  as  will  be  shown  by  some  simple  equations.  We  know 
that  when  a  body  moves  at  uniform  rate  in  a  circle,  the  force 
toward  the  center  must  be  equal  to  mv2/r,  m  being  the  mass, 
v  the  speed,  r  the  radius.  If  we  let  ^  represent  the  frequency 
of  revolution,  v  =  2-*^  and  the  force  may  be  written  47r2mr7?2. 
If  we  put  this  force  proportional  to  the  distance, 

4^2mr^2  =  Cr    or    ^  = 

showing  that  the  frequency  is  independent  of  the  distance. 
That  is  the  electron  would  make  the  same  number  of  revolu- 
tions per  second  whether  it  were  far  from  or  close  to  the  nu- 
cleus, and  if  these  revolutions  sent  out  light  waves,  the  latter 
would  have  the  same  length  whether  the  revolutions  of  the 
electron  were  violent  (large  radius  of  orbit)  or  weak.  On  the 
other  hand,  suppose  that  the  central  force  is  the  usual  inverse 
square  force  of  electrostatic  attractions.  Then 


r^ 

E,  and  e  being  respectively  the  charges  of  the  nucleus  and  the 
electron.  Here  the  frequency  depends  on  the  radius  r,  and 
smaller  orbits  would  be  traversed  with  higher  frequencies  than 
larger  ones,  and  therefore  send  out  shorter  waves. 

If  an  electron  gives  out  waves  as  a  result  of  its  rotation, 
it  is  bound  to  lose  energy,  and  therefore  to  draw  closer  to  the 
nucleus.  If  the  central  force  is  directly  proportional  to  the 
distance,  this  would  make  no  difference  in  the  wavelength 
emitted,  and  sharp  spectrum  lines  would  be  the  result.  On 
the  other  hand,  if  the  force  is  inversely  as  the  distance  squared, 


282 


LIGHT 


the  electron  would  give  out'  shorter  and  shorter  waves  as  it 
lost  energy,  so  that  sharp  spectrum  lines  would  not  result. 
Since  gases  emit  spectrum  lines  which  are  usually  more  or  less 
sharp,  often  so  sharp  that  only  the  most  refined  spectrum  ap- 
paratus can  show  that  they  are  not  absolutely  so,  this  evidence 
is  strongly  in  favor  of  a  central  force  proportional  to  the  dis- 
tance, and  therefore  of  a  spherical  nucleus  with  electrons  in- 
side. This  is  the  basis  of  the  atom  theory  proposed  at  one  time 
by  Sir  J.  J.  Thomson. 

But  certain  experiments  in  radioactivity,  and  some  other 
considerations,  indicate  that  the  nucleus,  at  least  in  some  ele- 
ments, is  much  too  small  to  contain  the  electrons,  and  this  fact 
has  made  the  Thomson  atom  somewhat  obsolete. 

Ritz's  theory  assumes  that  the  rotating  electrons  are  under 
the  control  of  a  magnetic  field  produced  by  the  nucleus,  which 
is  assumed  to  have  magnetic  properties.  By  introducing  cer- 
tain vibrations  of  the  nucleus  itself,  Bitz  was  able  to  explain 
the  variations  of  the  Zeeman  effect.  This  theory,  however,  has 
not  been  given  much  attention  by  physicists,  perhaps  because 
it  does  not  seem  to  agree  very  well  with  the  results  of  radio- 
activity experiments. 

155.  Bohr's  theory  of 
the  hydrogen  atom.— With- 
in recent  years  an  atom 
theory  has  been  formulated 
by  Bohr,  in  which  the  elec- 
tron is  considered  as  being 
under  an  electrostatic  force 
of  attraction  toward  the 
center  of  its  orbit,  varying 
inversely  as  the  square  of 
the  distance.  We  shall  con- 
fine our  attention  to  Bohr's 
application  of  the  theory  to 
the  simplest  element,  hydrogen.  He  conceives  the  single  elec- 
tron of  the  hydrogen  atom  as  capable  of  revolving  in  any  one 
of  a  large  number  of  orbits,  as  indicated  by  the  circles  1,  2,  3, 
etc.,  in  figure  137,  and  that  during  such  rotation  it  neither  radi- 
ates nor  absorbs  energy.  The  frequencies  of  revolution  in  these 


Figure    137 


BOHR'S  THEORY  28:* 

orbits  are  governed  by  the  equation  (2)  above,  which  for  this 
case  may  be  written 

since  for  hydrogen  E  =  e.  The  electron  would  have  a  different 
energy  as  well  as  a  different  revolution  frequency  in  each  orbit. 
The  potential  energy  of  a  negative  charge  e  at  a,  distance  r 
from  a  positive  charge  e  is  C  -  -  e2/r,  where  C  is  a  constant, 
the  potential  energy  when  the  electron  is  removed  to  an  infinite 
distance  from  the  nucleus.  The  kinetic  energy  is  of  course 
mv2/2,  or  27r2mr2^2,  since  v  =  2?rr^.  If  we  substitute  the  value 
of  ^  from  (2),  the  kinetic  energy  becomes  e2/2r.  This  makes 
the  total  energy 


Thus  orbits  of  smaller  radius  have  smaller  energy,  and  higher 
revolution  frequency. 

Now  it  has  long  been  suspected  that  an  atom  cannot  absorb 
or  emit  light  when  in  the  normal  condition,  but  that  absorp- 
tion occurs  when  it  is  being  ionized,  and  radiation  when  the 
electrons  are  in  the  act  of  recombining.  Accordingly,  Bohr 
supposes  that  the  hydrogen  electron,  in  jumping  from  an  orbit 
of  larger  to  one  of  smaller  radius,  to  do  which  it  must  get  rid 
of  a  certain  amount  of  energy,  emits  this  energy  in  one,  two, 
or  an  integral  number  of  quanta.  If  the  electron  jumps  from 
what  is  practically  an  infinite  distance  (complete  ionization) 
to  the  innermost  ring,  it  emits  one  quantum,  if  to  ring  2  two 
quanta,  to  ring  3  three  quanta,  etc.  The  frequency  of  the 
emitted  light,  which  we  shall  call  v,  is  not  the  same  as  the 
revolution  frequency  ^  of  the  orbit  at  which  it  chances  to  stop. 
Bohr  assumes  that  it  is  half  of  ^  and  this  assumption  fixes  the 
radii  of  the  orbits.  For,  if  we  call  rn  the  radius  of  the  nth  orbit. 
the  energy  in  that  orbit  is  C  —  e2/2rn  while  the  energy  at  an 
infinite  distance  is  simply  C.  Therefore  the  radiated  energy 
is  C  __  [C  —  e2/rD]  =  e2/2rn.  Since  this  is  equal  to  n  quanta, 
or  n  times  hv,  and  v  is  ^/2,  we  have 

e2/2rn  =  nhv  =  nh^/2 


284  LIGHT 

.Recalling  the  value  of  r/  given  in  equation  (2),  we  get,  by  sim- 
ple substitution  and  solution, 

ru  =  n2h2/47T2e2m  (4) 

If  we  substitute  this  value  of  rn  for  r  in  equation  (3),  we  get 
for  the  energy  in  the  nth  orbit 

Wu  =  C  —  27r2e4m/n2h2  (5) 

Now  suppose  that  the  electron,  instead  of  being  at  the  start 
completely  removed  from  the  neighborhood  of  its  nucleus,  is 
only  in  one  of  the  outer  orbits,  and  it  jumps  to  an  inner  one. 
Let  us  say  that  it  goes  from  the  nth  to  the  n'th  orbit.  The  en- 
ergy radiated  will  be 

Wn  —  W01  =  C  —  27r2e4m/n2h2  —  [C  —  27r2e4m/n'2h2] 


27r2e4m  /  J._       JL\ 
T2""  Vn72""^/ 


Bohr  assumes  that  in  this  case  the  energy  is  emitted  in  a  sin- 
gle quantum,  so  that  the  frequency  of  the  emitted  radiation  can 
be  gotten  by  dividing  this  radiated  energy  by  h,  giving 


This  can  be  put  into  terms  of  the  wavelength  A,  instead  of  the 
frequency  v,  by  substituting  A  =  c/v,  c  being  the  velocity  of 
light.  This  gives 

ch*      /     n2n'2     \ 
"2^e^  U2  —  n'2/ 

If  we  calculate  the  numerical  value  of  the  fraction  outside  the 
parenthesis,  we  must  put  c  =  3  X  1010,  h  =  6.55  X  10'27,  e  = 
4.78  X  30-10,  m  =  8.9  X  10"28.  This  gives 

A-  909X10-*^^  (8) 

Now  consider  the  array  of  wavelengths  that  would  be  ob- 
tained when  an  electron  jumps  to  the  first  orbit  from  the 
second,  then  from  the  third,  then  from  the  fourth,  etc.  For 
this  purpose,  we  put  n'  =  1  throughout,  and  let  n  take  the 
successive  values  2,  3,  4,  5,  etc.  This  will  give  a  spectral  series, 
of  formula 


BOHR'S  THEORY  285 


This  formula  about  corresponds  to  a  series  the  beginnings  of 
which  were  found  by  Lyman  in  the  far  ultraviolet.  It  also 
comes  at  about  the  right  place  for  the  K-series  of  X-rays  for 
hydrogen. 

If  we  consider  the  wavelengths  gotten  when  an  electron 
jumps  to  the  second  orbit,  from  the  third,  then  from  the  fourth, 
etc.,  the  series  formula  will  be,  putting  n'  =  2, 


n2  — 4 
=  3636  X  10 f 


n2  — 4 

Notice  that  this  is  the  form  of  the  Balmer  formula,  for  the 
visible  series  of  hydrogen  lines,  found  by  actually  considering 
the  wavelengths  of  the  lines  as  found  experimentally  (see  chap- 
ter VII),  even  the  value  of  the  numerical  constant  being  as 
nearly  the  same  as  we  could  expect,  considering  uncertainties 
in  the  values  of  h,  e,  and  m. 

If  we  let  n'  equal  3,  or  4,  etc.,  while  n  takes  all  the  integral 
values  larger  than  n',  we  get  other  possible  series,  according  to 
Bohr's  theory.  Traces  of  one  such  series,  that  for  n'  =  3,  have 
been  found  in  the  infrared.  Wavelengths  in  the  other  series 
would  come  very  far  in  the  infrared,  and  are  not  known  ex- 
perimentally. 

Bohr  does  not  attempt  to  explain  why  the  electron  does 
not  radiate  when  revolving  in  the  fixed  orbits,  as  it  should  do 
according  to  the  known  laws  of  electromagnetic  phenomena,  nor 
in  what  manner  the  radiation  occurs  in  the  jump  from  one  orbit 
to  another.  Still,  it  is  almost  inevitable  that  a  theory  which 
makes  use  of  the  quantum  theory  should  do  violence  to  the  laws 
of  mechanics  or  electromagnetics,  and  it  is  certainly  remark- 
able that  he  could  predict  so  closely  the  wavelengths  of  the 
hydrogen  series  by  making  use  of  values  of  h,  e,  and  m  found 
from  entirely  distinct  experiments.  At  any  rate,  the  theory  is 
regarded  seriously  by  physicists,  as  a  beginning  toward  an 
atom-theory  that  admits  of  quantitative  predictions  which  can 
be  compared  with  experimental  data. 


LIGHT 


APPENDIX  1. 

It  is  often  necessary,  in  optical  experiments,  to  have  at 
command  some  means  of  producing  bright-line  spectra  of  known 
wavelengths.  A  number  of  different  sources  that  yield  such 
spectra  are  mentioned  below,  and  the  principal  lines  given. 
The  wavelengths  are  expressed,  not  in  centimeters,  but  in  the 
unit  most  often  employed  for  this  purpose,  the  angstrom,  which 
is  equal  to  1O8  cm. 

The  easiest  source  of  all  to  work  with  is  the  sodium  flame 
spectrum.  The  two  bright  yellow  lines,  5890  and  5896,  are 
close  enough  together  so  that  for  many  purposes  they  answer 
well  enough  for  truly  monochromatic  light.  They  are  obtained 
with  considerable  brightness  by  soaking  a  thin  strip  of  asbestos 
paper  in  a  solution  of  common  salt,  and  tying  it  about  the 
mouth  of  a  Bunsen  burner,  so  that  the  flame  burns  from  the 
top  edge  of  the  asbestos.  If  lithium  chloride  be  used  instead 
of  common  salt,  a  bright  red  line  is  given,  of  wavelength  6708, 
but  usually  there  is  enough  sodium  present  as  an  impurity  to 
show  its  lines  also.  The  lithium  line  is  not  as  bright  as  the 
sodium  lines,  nor  so  persistent.  Quarter-wave  plates  are  usually 
adjusted  for  the  sodium  lines,  and  this  light  is  also '  used  in 
measuring  the  rotating  power  of  sugar  solutions. 

A  very  bright  and  convenient  source  is  the  mercurj^  arc. 
It  is  made  in  various  forms,  but  each  consists  of  a  glass  or 
quartz  tube  partly  filled  with  mercury,  the  air  being  pumped 
out.-  When  a  current  is  once  started  through  this,  it  continues 
to  run  through  mercury  vapor,  giving  a  number  of  bright  lines. 
The  principal  wavelengths  are  5790  and  5768,  both  yellow  and 
bright,— 5461,  yellow-green  and  very  bright, — 4916,  blue-green, 
weak, — 4359,  blue,  strong, — 4047,  violet,  strong.  The  line  5461 
is  probably  the  strongest  line  available. 

Hydrogen,  at  a  few  millimeters  pressure  in  a  glass  tube, 
and  excited  by  a  high-tension  electrical  discharge,  shows  a  great 
number  of  not  very  strong  lines,  but  also  the  following  series 
lines:  6563  red,  4861  blue-green,  4341  blue.  The  first  of  these 
is  the  strongest,  the  last  the  weakest.  The  next  line  of  the 

(286) 


VELOCITY  OF  LIGHT  287 

series,  4102,  does  not  show  visibly,  because  it  takes  a  fairly 
strong  line  to  show  to  the  eye  in  the  violet. 

Under  the  same  conditions,  helium  gives  a  number  of  clear 
bright  lines,  including  the  following  wavelengths:  6678  red, 
5876  yellow,  5048,  5016,  and  4922,  green,  4713  greenish  blue, 
4472  blue,  4388  violet.  The  yellow  line  is  very  strong,  the  lines 
5016,  4472,  and  4388  rather  strong,  the  rest  weaker. 

A  good,  bright,  and  practically  monochromatic  beam  can 
be  obtained  by  passing  the  light  from  the  mercury  arc  through 
a  lens  and  a  prism,  so  as  to  form  the  spectrum,  and  allowing 
one  of  the  brightest  lines  alone  to  pass  through  a  slit  in  the 
plane  in  which  the  spectrum  is  focussed.  The  interference 
rings  of  figure  84  A  were  photographed  with  light  obtained  in 
this  manner. 

APPENDIX  II. 

The  following  may  serve  as  a  proof  that,  for  the  special 
case  of  plane  monochromatic  waves,  the  velocity  of  propagation 

is  l/\/V 

According  to  Faraday's  theory  of  di-  ^<- 

electrics,  if  a  charge  4-  e  be  given  to  any 
conducting  sphere,    (figure    138)    an   equal      / 
amount    of   electricity    is    pushed    outward      I  vS^ 

\  i 

through  every  surface  surrounding  it.  If  we       \ 
take  the  surface  of  a  sphere  of  radius  r,  the 
amount  pushed  through  each  square  centi-  Figure  us 

meter  will  be  e/47r2,  and  this  amount  is  defined  as  the  ' '  displace- 
ment" and  is  represented  by  D.  The  intensity  of  the  electric 
force  at  the  surface  of  this  same  sphere  is  e/kr2.  Hence,  for 
this  case, 

D  =  kF/4*- 

and  it  can  be  shown  that  this  relation  holds  in  all  cases.  Fara- 
day regarded  D  as  analogous  to  a  mechanical  "strain,"  F  to 
a  mechanical  "stress,"  and  therefore  47r/k  is  analogous  to  an 
elastic  coefficient. 

Consider  a  system  of  three  orthogonal  axes,  as  in  figure 
139,  and  suppose  that  a  train  of  plane  monochromatic  waves 
are  advancing  along  the  positive  direction  of  the  X-axis.  We 


LIGHT 


shall  suppose  that  they  are  polarized  so  that  the  electrical 
vibrations  are  along  the  Y-axis,  the  magnetic  along  the  Z-axis. 

If  we  represent  the  former  by  F,  the 
c  *  latter  by  H,  each  must  follow  an  equa- 

tion of  the  type  of  (4)  in  chapter  III. 

F  =  K.cos~  (x  — Vt) 
x  A 

2-r 
Figure  139  H  =  M.  COS  -j-   (x  —  Vt) 

where  K  and  M  are  the  amplitudes  respectively  of  the  electrical 
and  the  magnetic  vibrations. 

In  order  to  find  the  relation  between  V,  /*,  and  k,  we  must 
apply  the  two  fundamental  laws  of  electromagnetics. 

A.  First  Law.  Consider  a  very  narrow  rectangle  abdc, 
perpendicular  to  the  Y-axis,  which  extends  in  width  from  x 
to  x  +  zJx,  but  may  have  any  length,  say  L.  The  amount  of 
electricity  which,  at  any  instant  t,  has  been  pushed  through  the 
area,  is  gotten  by  multiplying  the  area  L/jx  by  the  value  of  D 
in  this  neighborhood.  This  gives 

kL  2?r 

E  =    DL/lx  =kFLjx/47r  —  —   /IxK.  cos    —  (x  —  Vt) 

4<7T  A 

Since  E  is  changing  with  the  time,  this  constitutes  a  current, 
whose  value  is  the  limit  of  zlE/zlt.  The  value  of  E  at  time 
t  4-  At  is 


E  4.  JE  =         /jxK.  cos  -  (x  —  Vt  —  V/jt) 


,  .  , 

cos  —  (x  —  Vt)  +  sin  -  sm  —  (x  —  Vt) 


A 

The  last  equation  is  gotten  by  putting 
cos 

l^UO 

which  is  permissible  since  At  becomes  vanishingly  small  in  the 
limit. 


VELOCITY  OF  LIGHT  289 

If  we  now  subtract  the  value  of  E  for  time  t,  and  divide  by 
we  get  the  current  equal  to 


According  to  the,  laws  and  definitions  of  electromagnetic 
phenomena.  4?r  times  the  current  is  equal  to  the  work  done  in 
carrying  a  unit  north  pole  about  this  area,  in  the  direction 
abdca.  The  work  done  in  going  from  a  to  b  is  exactly  equal 
and  opposite  to  that  from  d  to  c,  but  that  from  b  to  d  is  not 
necessarily  balanced  by  that  from  c  to  a,  since  the  magnetic 
force  at  distance  x  +  /Jx  from  the  origin  is  in  general  different 
from  that  at  distance  x.  Evidently  the  net  work  done  in  the 
path  is  the  distance  bd  =  ac  =  L,  multiplied  by  the  excess  of 
the  value  of  H  at  distance  x  from  the  origin  over  that  at  dis- 
tance x  -f-  zlx,  that  is  by  —  zlH.  At  distance  x, 


and  at  distance  x 


~?  (x  —  Vt) 

A 


=  M.cos—  (x—  -Vt-h/Jx) 
A 


=  M  f  co.s  ^  (x  -  Vt)  -^-  sin  ^-  (x  -  Vt)  1 

L  A  A  A  J 

as  can  be  proved  by  trigonometrical  transformations  similar  to 
those  carried  out  above.     This  gives 

^TT  STT/JX  .      2-7T  . 

zJH  =  --  -  —  M.  sm  —  (x  —  Vt) 
A  A 

and  the  work  done  in  carrying  the  unit  pole  about  the  rectangle 


,rT     .    2-jr 

ML.  sin  —  (x  —  Vt) 


A  A 

Putting  this  equal  to  4rr  times  the  current,  we  get 


ML.  sin       (x  -  Vt)  =  4,  KV.  sin        (x  -  Vt  ) 

A  A  Z\  A 

M  =  kKV 


290  LIGHT 


kV  (1) 

B.  Second  Law.  The  second  law  states  that  the  electro- 
motive force  induced  in  any  circuit  is  equal  to  the  rate  of 
change  in  the  number  of  lines  of  force  through  thals  circuit, 
and  is  in  such  a  direction  as  to  oppose  that  change. 

Take  this  time  a  long  narrow  rectangle  perpendicular  to 
the  Z-axis.  The  number  of  lines  of  force  through  it  is  ju,  times 
H  times  the  area.  If  the  length  of  the  rectangle  is  again  L 
and  its  width  zJx,  this  number  is 

/xHL/jx  =  /xLjxM.  cos  -      (x  —  Vt) 
A 

The  rate  at  which  this  changes  with  the  time  is  easily  found, 
by  the  methods  used  above,  to  be 

^LzlxMV  —  sin—  (x  —  Vt) 

•\  A 

The  electromotive  force  is,  by  definition,  the  work  done  in 
carrying  a  unit  +  charge  about  the  circuit.  By  the  same 
process  already  used  in  finding  the  work  done  in  carrying  a 
unit  pole  about  the  other  circuit,  we  find  the  electromotive 
force  to  be 


,^T      .  , 

—  -  KL.  sin  —  (x  —  Vt) 
A  A 

Since  this  opposes  the  increase  in  the  lines  of  force,  we  have 
KL.  sin  -    (x  -  Vt)  =  jiLJxMV—  sin  -  (s  -  Vt) 


. 

A  A  A  A 


K  = 

M/K  —  1/>V  (2) 

Now,  if  we  eliminate  M/K  between  equations  (1)  and  (2), 
we  get  the  required  relation 


INDEX. 


References   are   to   pages. 


aberration,  chromatic,  83 
spherical,  91 

stellar,   6,   14 

absolute  motion,  136,  174 
absorption,  21,  23,  117,  118, 

of  X-rays,  271 
acceleration  in  s.  h.  m.,  180 
accommodation,  27 
achromatic  lens,  84 
amplitude,  40,  175 
analyzer,  226 
angle  of  incidence,  52 

polarization,   219 

reflection,  52 

refraction,  52 

anomalous    dispersion,    257 
Arago,   9 
arc,  113 

astigmatism,  28,  93 
atomic  numbers,  270,  280 
atoms,  264,  269,  279 

Balmer,  115,  285 
Baly,  115 

bending  into  shadow,  34 

biprism,  152 

black,  2,  23 

black  body,   119 
Bohr,  282 

bolometer,  132 
Bradley,  4,  5 
Bragg,  266 

bright-line  spectra,  111 

brightness  of  images,  202 
Bunsen  photometer,  197 

calcite,  206 

camera  lenses,  94 

Canada  balsam,  224 
candle-power,   199 


257 


cathode  rays,  262 
Cauchy,  254 
caustic,  92 
circular  motion,  192,  231 

polarization,    228,    237 
clouds,  whiteness  of,  24 
collimator,    110 
color,  1,  15-30, 

blindness,  29,  30 

mixture,  25 

vision,  28 
colors,  impure,  19 

unsaturated,  20 
complementary  colors,  24 
concave  grating,  129 
confocal  planes,  88 
conjugate  foci,  74 
continuous  spectrum,  116,  119 
corpuscular  theory,  32,  273 
critical  angle,  55 
crystal  reflection  of  X-rays,  266 

structure,  268 
crystals,  biaxial,  217 

uniaxial,  206 
curvature  of  field,  92 

dark-line  spectra,  18,  117 

defects  of  mirrors  and  lenses,  91 

density,  optical,  54 

deviation,  60 

diamond,  54,  57 

dielectric    constant,    250,    252, 

256 

diffraction,  91,  99,  142-151,  263 
diopter,  83 

dioptric   strength,   83 
dispersion,  110,  254,  255 

anomalous,    257 
displacement  in  s.  h.  m.,  175 
displacement-currents,  245 


291 


292 


LIGHT 


displacement,  electrical,  287 
distance   of   distinct  vision,   27, 

104 
Doppler  effect,  133 

double-image  prism,  225 
double  refraction,  206 

eclipse,  4 
Einstein,  174 

elastic-solid  theories,  240,  273 
electromagnetic  laws,  245 

theory,  241,  273 

waves,  242 

electron  theory,  255,  276 
elliptic   polarization,    226,   237 

s.  h.  m.,  190 
energy  in  s.  h.  m.,  182 
ether,  33,  36,  135,  136,  171 
exit-pupil  99 
extraordinary  ray  207 
eye,  26 
eye-lens,  100 
eye-piece,  100 

Faraday,  233,  245,  287 

field-lens,  100 

field  of  view,  99 

fish-eye  vision,  58 
Fitzgerald,  173 
Fizeau,  4,  7,  9 

fluorescence,  130,  263 

focal  length  of  a  lens,  12,  80,  84 
mirror,  74 
plane,  principal,  89 

foci,  conjugate,  11,  74,  80 

focus,  principal,  74  80 

foot-candle,  159 
Foucault,  4,  9,  35 
Fourier,  38 

Fresnel,   44,   152,   231, 
Friedrich,  266 

fringes,  43 

Galileo,  3 

grating,   concave,  129 
plane,  121 
reflection,  129 


half-silvered  mirror,  8,  10,  160 

half-wave  plate,  216 
Hallwachs,  278 

heat-waves,  131 
Hefner  lamp,  199 
Hering  theory,  30 
Hertz,  247 
Humphreys,    170 
Huyghens,  51,  139,  206 
eyepiece,  106 
zones,  139 

hydrogen  spectrum,  114,  282 

hypermetropic  eye,  27 

Iceland  spar,  206 
image  by  reflection,  64 

refraction,   65 

of  extended  object,  69,  87 

real,  71 

virtual,    71 
imperfections    of    mirrors    and 

lenses,  91 

impressionist  painting,  26 
impure  colors,   19 
index  of  refraction,  53,  252,  254 
infrared,  131 
intensity  of  light,  195 
interference,  42,  152,  187 

fringes,    43 

in  white  light,  48,  158,   163 
interferometer,  160,  163 
intrinsic  luminosity,  200 
inverse  square  law,  195 
ionization  by  X-rays,  263 

judgment  of  distance,  67 
Jupiter,  4 

K-series  of  X-rays,   270 
Kayser,   115 
Kerr,   233 
Knipping,  266 

L-series  of  X-rays,  270 

lantern,  opaque  projection,  108 

simple  projection,  107 
Laue,   266 

lens,  77 


INDEX 


293 


achromatic,  84 
formula  for,  79,  80 
lenses,  two  in  contact,  82 

types  of,  81 
light-standards,  199 
light-year,  14 
Lissajous  figures,  193 

longitudinal  waves,  37,  205 
Lorentz,  173,  276 

luminous  objects,  1 
Lummer-Brodhun  photometer,  198 

magnetic  rotation,  233 
magnification,  88 
magnifying  power,  98 
Marx,   264 
Maxwell,  244,  247 
Michelson,  13,   160,  171 
micrometer,   12,   90 
miscroscope  compound,  105 
objective,  95 
simple,  104 

minimum  deviation,  61 
mirror,  formula  for  curved,  74 
half-silvered,  8,  10,  160 
rotating,  10 
mirrors,   Fresnel,    44 
molecules,  23^  269 
monochromatic  light,  48 
Morley,    171, 
myopic  eye,  27 

Newton,  15,  33,  35, 
Newton's  rings,  162 
Nicol  prism,  224 

occultation,   14 

opaque  projection  lantern,  108 

opera-glass,   101 

optic  axis,  207 

optical   rotation,    230 

ordinary  ray,  207 


parallax,  6,  14,  69 
parallel  s.  h.  m/s,  184 
parsec,  14 
period,  4,  175 
permeability,  250 


phase,  38,  40,  42,  175 
phase-change  on  reflection,  155 
phase-constant,  40 
phosphorescence,  131 
photo-electric  effect,  278 
photography,  131 
photometers,  196 
photometry,  196 
pigments,  20,  25 
Planck,  120 

plane  grating,  121 

of  polarization,  211,  240,  243 

polarized  light,  211,  273 

waves,  37,  50 
polarization  by  reflection,  218 

electrical,  256 

polarized  light,  plane,  211,  273 
polarizer,  226 
presbyopia,  28 
principal  focal  plane,  89 

focus,  74,  80 

plane,  208 
prism,  15-26,  59-61 

double-image,  225 
prism-binocular,    103 
prism-spectroscope,   110 
projection  lantern,  107 
propagation    of    electromagnetic 
waves,  248 

quantum   theory,   120,   271,    274, 

279,    283 
quarter-wave  plate,  216 

rainbows,   22,   167-170 
Ramsden  eyepiece,   100 
ray,  extraordinary,  207 
ordinary,   207 
undeviated,  88 
ray-velocity,  215 
real  image,  71 

spectrum,  17 
rectangular  opening,  diffraction 

through,  145 
rectilinear  propagation,  2,  34, 

13S 

reflecting  telescope,  103 
reflection,  2,  32,  50,  63,  69 


294 


LIGHT 


at  plane  mirror,  50,  63 
spherical  surface,  69 
grating,  129 

reflection  of  polarized  light,  219 
X-rays,   264 
total,  54 
reflector,    103 
refraction,  2,  15,  32,  35, 

at  a  plane  surface,  50,  63 
spherical  surface,  69 
refractive  index,  53,  252,  254 
relativity  theory,  173,  274 
residual  rays,   259 
resolution  of  circular  motion,  231 

s.  h.  m.,  192 
resolving-power,  148 
reststrahlen,  259 
retina,  27,  28 
rings  and  brushes,  234 
Ritz,  278,  282 
Rcchon  prism,  225 
Roentgen,  262 

rotation  of  plane  of  polarization, 

230,  23a 
round  opening,   diffraction 

through,  149 
Rowland,  129 
Rumford  photometer,  196 
Runge,  115 
Rydberg,  115 

sagitta  of  an  arc,  65 
satellites  of  Jupiter,  4 
scattering  of  light,  22 
secondary  X-rays,  271 
series  in  spectra,  114 

X-rays,   270 
shadow  of  an  edge,  34,  142 

a  wire,  145 
simple  harmonic  motion,  175-193 

miscroscope,   104 
sky,  blue  of,  22 
slit,  10,  16,  19,  42 
snow,  whiteness  of,  23 
solar  spectrum,  18,  117 
solid  angle,  199 
source  of  light,  42 
spectral    series,    114 


spectrograph,   111 
spectrometer,  111,  204 
spectrophotometer,  204 
spectroscope,  110 
spectrum,  17,  111,  116,  117 

lines,    112 

of  X-rays,  270 
spherical  aberration,  91 
standards  of  light,  199 
Stark  effect,  278 

telescope,   97,   110 

reflecting,  103 

lenses,  94, 
theories  of  color  vision,  28 

light,  32 
thermopile,    132 
thin  films,  interference  in,  153, 

159. 

Thomson,  282 
toothed  wheel,  7 
tourmaline  216 
transparent  media,  2 

ultramicroscope,    107 
ultraviolet,  130 
unsaturated  colors,  20 

vacuum-tube,  113 

velocity   in    s.   h.   m.,    178 

velocity  of  light,  2-ia,  35,  52,  244, 

250,  287 
Vernon-Harcourt   lamp,  199 

vibrations  in  polarized  light,  243 
virtual    spectrum,   17 
image,  71 

wave,  23,  33,  37,  40 
theory,  32  273 
wavefront,   38 

wavelength,  36,  40,  47,  260,  272, 
286 

of  X-rays,  264,  267,  269 
wave-surface  in  biaxial  crystals, 
217 

uniaxial  213 

Wollaston  prism,  225 
Wood,  58  170 


INDEX 


295 


X-rays,  262 

secondary,  271 
series  in,  270 
velocity  of  264 
wavelengths  of,  264,  267,  269 


Young-Helmholtz    theory,    29 
Young's    interference    experiment, 
152 

Zeeman  effect,  275 


14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 
Renewed  books  are  subject  to  immediate  recall. 


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LOAN  DEPT. 





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